OF  THE 

University  of  California. 

Col.  t.   h    '^\^D3^jRr. 

Class 

ELEMENTS 


QUATEKNIONS. 


A.    S.    HARDY,    Ph.D., 

PROFESSOR   OF   MATHEMATICS,    DARTMOUTH    COLLEGE. 


Of  THE     ^ 


BOSTON: 

PUBLISHED   BY   GINN,   HEATH,  &  CO. 
1881. 


Entered  according  to  Act  of  Congress,  in  the  year  1881,  by 

A.  S.  HARDY, 
in  the  office  of  the  Librarian  of  Congress,  at  Washington. 


GiNN,  Heath,  &  Co. 

J.  S.  Gushing,  Printer,  i6  Hawley  Street, 

Boston. 


V 


or  THE 

^NivERsiry 


or 


PREFACE. 

n^HE  object  of  the  following  treatise  is  to  exhibit  the 
elementary  principles  and  notation  of  the  Quaternion 
Calculus,  so  as  to  meet  the  wants  of  beginners  in  the 
class-room.  The  Ulements  and  Lectures  of  Sir  AVilliam 
Rowan  Hamilton,  while  they  may  be  said  to  contain  the 
suggestion  of  all  that  will  be  done  in  the  way  of  Quater- 
nion research  and  application,  are  not,  for  this  reason,  as 
also  on  account  of  their  diffuseness  of  style,  suitable  for 
the  pur])oses  of  elementary  instruction.  Tait's  work  on 
Quaternions  is  also,  in  its  originality  and  conciseness, 
beyond  the  time  and  needs  of  the  beginner.  In  addition 
to  the  above,  the  following  works  have  been  consulted: 

Calcolo  del  Quaternione.     Bellavitis ;  Modena,  1858. 

Exposition  de  la  Metliode  des  iquipollences.  Traduit 
de  ritalien  de  Giusto  Bellavitis,  par  C.-A.  Laisant ;  Paris, 
1874.  (Original  memoir  in  the  Memoirs  of  the  Italian 
Society.     1854.) 

Theorie  Elementaire  des  Quantites  Complexes.  J. 
Hoiiel;   Paris,  1874. 

Essai  sur  une  ManiP.re  de  Representer  les  Quantites 
Imaginaires  dans  les  Construction  Geometriques.  Par 
R.  Argand ;    Paris,  1806.     Second  edition,  with  preface 


IV  PEEFACE. 

by  J.  Hoiiel ;  Paris,  1874.  Translated,  with  notes,  from 
the  French,  by  A.  S.  Hardy.  Van  Nostrand's  Science 
Series,  No.  62;    1881. 

Kurze  Anleitung  zum  Rechnen  mit  deM  (^Hamilton  scheti) 
Quaternionen.     J.  Odstrcil ;    Halle,  1879. 

ApjjUcations  Mecaniques  du  Calcul  des  Quaternions. 
Laisant ;   Paris,  1877. 

Introduction  to  Quaternions.  Kelland  and  Tait;  Lon- 
don, 1873. 

A  free  use  has  been  made  of  the  examples  and  exercises 
of  the  last  work ;  and,  in  Article  87,  is  given,  by  permis- 
sion, the  substance  of  a  paper  from  Volume  I.,  page  379, 
America7i  Journal  of  Mathematics,  illustrating  admirably 
the  simplicity  and  brevity  of  the  Quaternion  method. 

If  this  presentation  of  the  principles  shall  afford  the 
undergraduate  student  a  glimpse  of  this  elegant  and  pow- 
erful instrument  of  analytical  research,  or  lead  him  to 
follow  their  more  extended  application  in  the  works  above 
cited,  the  aim  of  this  treatise  will  have  been  accomplished. 

The  author  expresses  his  obligation  to  Mr.  T.  W.  D. 
Worthen  for  valuable  assistance  in  the  preparation  of 
this  work,  and  to  Mr.  J.  S.  Gushing  for  whatever  of 
typographical  excellence  it  possesses. 

A.  S.  HARDY. 
Hanover,  N.H.,  Juue  21,  1881. 


COXTEXTS. 


CHAPTER  I. 

Addition    and    Subtraction   of  Vectors ;    or,   Geometric 
Addition   and   Subtraction. 

Article.  Page. 

1.  Definition  of  a  rector.     Eflect  of  the  minus  sign  before  a 

vector  1 

2.  Equal  vectors 2 

3.  Unequal  vectors.     Vector  addition      2 

4.  Vector  addition,  commutative 3 

6.   Vector  addition,  associative      3 

6.  Transposition  of  terms  iu  a  vector  equation i 

7.  Definition  of  a  tensor i 

8.  Definition  of  a  scalar 5 

9.  Distributive  law  iu  the  multiplication  of  vector  bj^  scalar 

quantities G 

10.  If  i«  + ^,3  =  0,  then  2a  =  0  and  i;;J  =  0 7 

11.  Examples 8 

12.  C'omplauar  vectors.     Condition  of  complanarity 15 

13.  Co-initial  vectors.     Condition  of  collinearity      IG 

14.  Examples 17 

15.  Expression  for  a  medial  vector 2-t 

16.  Expression  for  an  angle-bisector 25 

17.  Examples 20 

18.  Mean  point      28 

19.  Examples 28 

20.  Exercises 30 


CHAPTER   II. 

Mnltiplication  and  Division  of   Vectors ;    or,   Geometric 
Multiplication  and  Division. 


21.    Elements  of  a  quaternion 


.>.> 


E(iual  quaternions 34 


23.   Positive  rotation 


VI  CONTENTS. 

Article.  Page. 

24.  Analytical  expression  for  a  quaternion.     Product  and  quo- 

tient of  rectangular  unit-vectors.     Tensor  and  vemur  of  a 

quaternion 36 

25.  Symbolic  notation  g  =  TgUg 39 

26.  Keciprocal  of  a  quaternion 39 

27.  Quadrantal  versors,  i,  j,  k     40 

28.  Whole  powers  of  unit  vectors.     Square  of  a  unit  vector  is  —  1,  41 

29.  Associative  law  in  the  multiplication  of  rectangular  unit- 

vectors      42 

30.  Negative  sign  commutative  with  i,  j,  k 43 

31.  Commutative  law  not  true  for  the  products  of  i,  j,  k   .   .    .   .  43 

32.  Reciprocal  of  a  unit  vector 44 

33.  A  unit  vector  commutative  with  its  reciprocal.     Reciprocal 

of  any  vector 45 

34.  Product  and  quotient  of  any  two  rectangular  vectors  ....  47 

35.  Square  of  any  vector 47 

36.  Distributive  law  true  of  the  products  of  i,  j.  k 48 

37.  Exercises 48 

38.  Symbolic  notation,  g  =  8(2 -f  Vg 49 

39.  De  Moivre's  theorem 50 

40.  Products  of  two  vectox'S.     Symbolic  notation 56 

41.  General  principles  and  formulae 58 

42.  Powers  of  vectors  and  quatei'uions (jl 

43.  Relation  between  the  vector  and  Cartesian  determination  of 

a  point G-1 

44.  Right,  coraplanar,  diplanar  and  collinear  quaternions.     Any 

two  quaternions  reducible  to  the  forms  -,  L  and  '1,  L  .   .  Go 

45.  Reciprocal  of  a  vector,  scalar  and  quaternion      6G 

46.  Conjugate  of  a  vector,  scalar  and  quaternion G8 

47.  Opposite  quaternions 70 

48.  UKg=U-  =  -^  =  KUg 71 

q      Vq 

49.  Representation  of  versors  by  spherical  arcs 71 

50.  Addition  and  subtraction  of  quaternions.     K,  S  and  V  dis- 

tributive symbols.     K  commutative  with  S  and  V.     T  and 

U  not  distributive  symbols 72 

51.  Multiplication  of  quaternious  ;  not  commutative.   lJuq  =  lllJq. 

Tnq=nTq.     Kgr=KrKr/.     Product  or  quotient  of  com- 

planar  quaternions 74 


Distributive  and  associative  laws  in  quaternion  and  vector 


52. 

multiplication 79 

63.   General  formulae 82 


CONTEXTS.  VU 

Article.  Page. 

54.  Applicatious 82 

55.  Formulae  relatiug  to  the  products  of  two  or  more  vectors  .    .  9!) 

56.  Exercises 100 

57.  Examples,     Applicatious  to  Spherical  Trigonometry    ....  108 

58.  General  Formulae 119 

69.  Applicatious  to  Plaue  Trigonometry 125 


CHAFTER  III. 
Applications    to    Loci. 

60.  General  equations  of  a  line  and  surftice 131 

61.  Use  of  Cartesian  forms  in  conjunction  with  quaternion  rea- 

soning   132 

62.  Non-commutative  law  in  quaternion  differentiation.     Differ- 

entiation of  scalar  functions 132 

63.  Quaternion  differentiation 134 

64.  Illustration 136 

65.  Distributive  principle 138 

66.  Quadrinomial  form 139 

67.  Examples 140 

The   Right  Line. 

68.  Right  line  through  the  origin 145 

69.  Parallel  lines 145 

70.  Right  line  through  two  given  points liC> 

71.  Perpendicular  to  a  given  line ;  its  length 147 

72.  Equations  of  a  right  line  are  linear  and  involve  one  indepen- 

dent variable  scalar 150 

The  Plane. 

73.  Equation  of  a  plane     151 

74.  Plane  making  eiiual  angles  with  three  given  lines      152 

75.  Plane  through  tlirce  given  points 1.53 

76.  Equations  of  a  plane  are  linear  and  involve  two  independent 

variable  scalars     154 

77.  Exercises  and  pi'obleras  on  the  right  line  and  plane 154 

The  Circle  and  Sphere. 

78.  Equations  of  the  circle 104 

79.  Equations  of  the  sphere 105 


Vlll  CONTENTS. 

Article.  Page. 

80.  Tangent  line  and  plane .  iqq 

81.  Chords  of  contact 167 

82.  Exercises  and  problems  on  the  circle  and  the  sphere    ....  167 

83.  Exercises  in  the  transfoi'iiuition  and  interpretation  of  ele- 

mentary symbolic  forms     176 

The   Conic  Sections.     Cartesian  Forms. 

84.  The  parabola 178 

85.  Tangent  to  tlie  parabola     178 

86.  Examples  on  the  parabola 180 

87.  Eelations  between  three  intersecting  tangents  to  the  para- 

bola    185 

88.  The  ellipse ,  191 

89.  Examples  on  the  ellipse 192 

90.  The  hyperbola I95 

91.  Examples  on  the  hyperbola    .    .    .   .    • 195 


92.  Linear  equation  in  quaternions.     Conjugate  and  self-conju- 

gate functions 199 

93.  General  equations  of  the  conic  sections 201 

94.  The  ellipse 204 

95.  Examples 206 

96.  The  parabola 214 

97.  Examples 216 

98.  The  cycloid 222 

99.  Elementary  applications  to  mechanics 223 


ELEMENTS  OF  QUATERNIONS. 


QUATERNIONS 


CHAPTER   I. 

Additiou  aud  Subtraction  of  A'octors,  or  Geometric  Addition  and 
Subtraction. 

1.  A  Vector  is  the  representative  of  transference  through  a 
given  distance  in  a  given  direction. 

Thus,  if  A,  B  are  any  two  points,  vector  ab  implies  a  trans- 
lation from  A  to  B. 

A  vector  maj'  be  represented  geometrically  by  a  right  line, 
whose  length  denotes  the  distance  over  which  transference  takes 
place,  and  whose  direction  denotes  the  direction  of  tlie  trans- 
ference. In  thus  designating  a  vector,  the  direction  is  indicated 
by  the  o)'cler  of  the  letters. 

Thus,  AB  (Fig.  1)  denotes  transference  i^?-i- 

from  A  to  B,  and  ba  from  t  to  a. 

Retaining  the  algebraic  signification  of  the  signs  +  .'^I'd  — .  if 
ab  denotes  motion  from  a  to  b.  then  — aij  will  denote  motion 
from  b  to  a,  and 

AB=— ba,        —  AB  =  BA  .        .        .       .       (1). 

Hence,  the  effect  of  a  miims  sign  before  a  vector  is  to  reverse 
its  direction. 

The  conception  of  a  vector,  therefore,  implies  that  of  its  two 
elements,  distance  and  direction ;  it  was  first  defined  as  a  directed 
right  line.  It  is  now  applied  more  generally  to  all  quantitiis 
determined  by  magnitude  and  direction.     Thus,  Ibrce,  the  path 


QUATERNIONS. 


of  a  moving  bod}',  A'elocity,  an  electric  current,  etc.,  are  vector 
quantities. 

Analytically',   vectors   are  represented  by  the  letters  of  the 
Greek  alphabet,  a,  /?,  y,  etc. 

2.    It  follows,  from  the  definition  of  a  vector,  that  all  lines 
which  are  equal  and  parallel  may  he  represented  by  the  same  vec- 
tor symbol  ivith  like  or  iinliJce  signs. 
If  equal  and  drawn  in  the  same 
direction,  they  will  have  the  same 
A.         Ry^        X^      F  sign.     Hence  an  equality  between 

\~      \.  two  vectors  implies  equality  in  dis- 

^ -^       tance  with  the  same  direction.      <" 

II  <j 

Thus,  if  AB  (Fig.  2),  cd,  be,  ef 
and  iiG  are  equal  and  drawn  in  the  same  direction,  they  may  be 
represented  by  the  same  vector  symbol,  and 

AB  =  CD  =  BE  =  EF  =  IIG  =  a     .       .       .       .       (2)  . 


Fig.  i 

c 


3.  It  follows  also  from  the  definition  of  a  vector  that,  if  vec- 
tors are  not  parallel,  they  cannot  be  represented  by  the  same 
vector  s^'mbol. 

Tluis,  if  the  point  a  (Fig.  3)  move  over  the  right  line  ab, 
from  A  to  B,  and  then  over  the  right  line  bc,  from  b  to  c,  and 
AB  =  a,   BC  must  be  denoted  by 
some  other  symbol,  as  /3. 

The  result  of  these  two  succes- 
sive translations  of  the  point  a  is 
the  same  as  that  of  the  single  and 
direct  translation  AC  =  y,  from  a  to 
c  ;  in  either  case  a  is  found  at  the 
extremity  of  the  diagonal  of  the 
parallelogram  of  which  ab  and  bc  are  the  sides.     This  combina- 
tion of  successive  translations  is  called  addition,  and  is  written 
in  tlie  ordinary  way,  ^  _|.  ^  ^  .^ (3), 

Tliis  expression  would  be  absurd  if  the  symbols  denoted  mag- 
nitudes only.     It  means  that  transference  from  a  to  b,  followed 


GEOMETRIC    ADDITION    AND    8UI5TP.A<'TI<  )N\ 


b}-  transference  from  b  to  c,  is  equivalent  to  transference  from 
A  to  c.  The  sign  +  does  not  tlierefore  denote  a  numerical  ad- 
dition, or  tlie  sign  =  an  equality  between  magnitudes.  It  is, 
however,  called  an  equation,  and  read,  as  usual,  "a  plus  fi  is 
equal  to  y."     This  kind  of  addition  is  called  geomdric  addition. 

4.  If  the  point  a  (Fig.  3) ,  instead  of  moving  over  the  sides 
AB,  BC  of  the  parallelogram  abcd,  had  moved  in  succession  over 
the  other  two  sides,  ad  and  DC,  the  result  would  still  have  been 
the  same  as  that  of  the  single  translation  over  the  diagonal  ac. 
But  since  ab  and  bc  are  equal  in  length  to  dc  and  ad  respect- 
ivel}',  and  are  drawn  in  the  same  direction,  we  have  (Art.  2) 

AB  =  DC     and     bc  =  ad, 
and  if  the  first  two  translations  are  represented  by  ab  and  bc, 
the  second  two  may  be  represented  b}-  bc  and  ab,  or 

a  +  /3  =  fS  +  a  =  y (O- 

Hence  the  operation  of  vector  addition  is  commutative,  or  tlio 
sum  of  any  ninnber  of  given  vectors  is  independent  of  their  ordrr. 

5.  If  the  point  a  (Fig.  4)  move  in  succession  over  the  three 
edges  AB,  BC,  cg  of  a  parallelopiped, 

Fig.  4. 


we  have 
and 

AB  +  BC  =  AC, 
AC  +  CO  =  AG, 

or 

(ab  +  Bc)  +  CG  =  AG. 

In  like 

manner 

BC  +  CG  =  BG, 

AB  +  BG  =  AG, 

or 

AB  +  (bC  +  Cg)  =  AG 

Hence 


(.^). 


(aB  +  BC)  +  CG  =  AB-f-(BC  +  CG)       . 

and  tJie  operation  of  vector  addition  is  associative,  or  the  sum 
of  any  number  of  given  vectors  is  independent  of  the  mode  of 
grouping  them. 


4  QUATERNIONS. 

6.  Since,  if  ac  =  y  (Fig.  3),  then  ca  =  — y,  we  have 

a  +  /3-y=0, 
or,  comparing  witli  equation  (3), 

a  +  ^  =  y, 

a  term  may  be  transposed  from  one  member  to  anotlier  in  a  vector 
equation  by  changing  its  sign. 

Also,  in  ever}'  triangle,  an}-  side  maj'  be  considered  as  the 
sum  or  difference  of  the  other  two,  depending  upon  their  direc- 
tions as  vectors.     Thus    (Fig.  3) 

y-/3  =  a, 

y-a  =  /3. 

It  is  to  be  observed  that  no  one  direction  is  assumed  as  posi- 
tive, as  in  Cartesian  Geometr}-.  The  onl}'  assumption  is  that 
opposite  directions  shall  liave  opposite  signs.  The  results  must, 
of  course,  be  interpreted  in  accordance  with  the  primitive  as- 
sumptions. Thus,  had  "we  assumed  BA  =  a  (Fig.  3),  y  and  /3 
being  as  before,  then 

a-/3=-y. 

7.  If  two  vectors  having  the  same  direction  be  added  together, 
•tlie  sum  will  be  a  vector  in  the  same  direction.  If  the  vectors 
be  also  c(pial  in  length,  the  length  of  the  vector  sum  will  be  the 
sum  of  their  lengths.  If  n  vectors,  of  equal  length  and  drawn 
in  the  same  direction,  be  added  together,  the  sum  will  be  the 
product  of  one  of  these  vectors  b}-  ??,  or  a  vector  having  the  same 
direction  and  whose  length  is  n  times  the  common  length.  If 
then  (Fig.  2) 

AF  =  a-AB  =  XCB  =  Xa, 

where  a,  b  and  v  are  in  the  same  straight  line,  CD  =  ab,  and  x 
is  a  positive  Avhole  number,  x  expresses  the  ratio  of  the  lengths 
of  AF  and  a.  From  the  case  in  which  x  is  an  integer  we  pass, 
by  the  usual  reasoning,  to  that  in  which  it  is  frnctional  or  in- 
commensurable. Vectors,  then,  in  the  same  direction,  have  the 
same  ratio  as  the  corresponding  lengths. 


GEOMETRIC    ADDITIOX   AND    SUBTRACTION.  5 

If  AB  =  a  be  assumed  as  the  unit  vector,  then 

AF  =  ma, 

in  which  m  is  a  ^^ositive  numerical  quantity  and  is  called  the 
Tensor.  It  is  the  ratio  of  the  length  of  the  vector  ma  to  that 
of  the  unit  vector  a,  or  the  numerical  factor  by  which  the  unit 
vector  is  multiplied  to  produce  the  given  vector. 

Any  vector,  as  ^8,  ma}-  be  written  in  general  notation 

/3  =  T/3r/3. 

In  this  notation,  T/?  (read  "tensor  of  /?")  is  the  numerical 
factor  which  stretches  the  unit  vector  so  that  it  shall  have  the 
I^roper  length ;  hence  its  name,  tensor.  It  is,  strictly  speaking, 
an  abstract  number  without  sign,  but,  to  distinguish  between  it 
and  the  negative  of  algebra,  it  maj'  be  said  to  be  always  posi- 
tive. l'/3  (read  "  versor  of  /?  ")  is  the  unit  vector  having  the 
direction  of  ^  ;  the  reason  for  the  name  versor  will  appear  later. 

T  and  U  are  also  general  symbols  of  operation.  Written  be- 
fore an  expression,  they  denote  the  operations  of  taking  the 
tensor  and  versor,  respectively.  Thus,  if  the  length  of  y8  is  n 
times  that  of  the  unit  vector, 

T(^)  =  n, 

where  T  denotes  the  operation  of  taking  the  stretching  factor. 
i.e.  the  tensor.     While 

r(/3)=r;8 

indicates  the  operation  of  taking  the  unit  vector,  that  is,  of 
reducing  a  vector  ^  to  its  unit  of  length  without  changing  its 
direction. 

8.   If  liC  (Fig.  5)  be  any  vector,  and  ra  =  ?/bc,  then 

—  BA  =  AB  =  —  ?/BC  ;  j,j„  5 

and.  in  general,  if  ba  and  bc  be      B c_ a 

any  two  real  vectors,  jiarallel  and 

of  unequal  length,  we  may  always  conceive  of  a  coeflicient  y 

which  shall  satisfy  the  equation 

BA  =  ^BC, 


6  QUATERNIONS. 

where  y  is  plus  or  minus,  according  as  the  vectors  have  the  same 
or  opposite  directions,  y  may  be  called  the  geometric  quotient, 
and  is  a  real  number,  plus  or  minus,  expressing  numericaJlj'  the 
ratio  of  the  vector  lengths.  This  quotient  of  parallel  vectors, 
which  may  be  positive  or  negative,  whole,  fractional  or  incom- 
mensurable, but  which  is  always  real,  is  called  a  Scalar,  because 
it  may  be  always  found  by  the  actual  comparison  of  the  parallel 
vectors  with  a  parallel  right  line  as  a  scale. 

It  is  to  be  observed  that  tensors  are  pure  numbers,  or  signless 
numbers,  operating  only  metrically  on  the  lengths  of  the  vectors 
of  which  they  are  coefficients :  while  scalars  are  sign-bearing 
numbers,  or  the  reals  of  Algebra,  and  are  combined  with  each 
other  by  the  ordinary  rules  of  Algebra ;  they  may  be  regarded 
as  the  product  of  tensors  and  the  signs  of  direction. 

Thus,  let 

a=  alia. 

Then  Ta  =  a.  If  we  increase  the  length  of  a  by  the  factor  &, 
h  is  a  tensor,  but  the  tensor  of  the  resulting  vector  is  h(i/^\i  we 
operate  with  —  6,  —  &  is  not  a  tensor,  for  a  is  not  onlj^tretched 
but  also  reversed  ;  the  tensor  of  the  resulting  vector  is  as  before 
ha  ;  in  other  words,  direction  does  not  enter  into  the  conception 
of  a  tensor.  As  the  product  of  a  sign  and  a  tensor,  —  6  is  a 
scalar.  The  operation  of  taking  the  scalar  terms  of  an  expres- 
sion is  indicated  by  the  symbol  S.  Thus,  if  c  be  any  i^eal  alge- 
braic quantity, 

S  ( —  &a  Ua  -f  c)  =  c, 

for  —  6a  Ua  is  a  vector,  and  the  onl}-  scalar  term  in  the  expres- 
sion is  c. 

9.  It  is  evident  from  Art.  7  that  if  a,  &,  c  are  scalar  coeffi- 
cients, and  a  any  vector,  we  have 

(a  4- Z> -f  c)  a  =  aa -I- 6a -f  ca  .     .     .     .     (G). 

Furthermore,  if  (Fig.  6) 

OA  =  a,      AB  =  /?,      BC  =  y,      Oa'  =  ?7la, 


GEOMETRIC    ADDITION   AND    SUBTRACTION, 
thou,   a'v.'  being  drawn  parallel  to  ai?  and  u'c'  to  uc, 
a'h'  =  m(3,     b'c'  =  my. 
oc  =  a  4-  /?  +  y,  ■^'«-  "• 

OC'  =  mOC  =  III  (a  +  /3  4-  y)  . 

But  wc  have  also 


Now 
and 


Hence 


oc'  =  ox'  +  a'h'  +  b'c' 
=  ma  +  ml3  +  my. 

m  (a  +  /3  +  y)  =  ')na  +  m^  +  7»,y 


('), 


or  i/((?  distributive  law  holds  good  for  the  muUijyIication  of  scala?' 
and  vector  quantities. 


10,    It  is  clear  that  while 

a  — a=0, 

a  ±  ^  cannot  be  zero,  since  no  amount  of  transference  in  a  direc- 
tion not  parallel  to  a  can  affect  a. 
Hence,  if 

na  +  m(i  =  0, 

since  a  and  /S  are  entirely  independent  of  each  other,  we  must 
have 

na  =  0     and     mj3  =  0, 
or 

01  =  0     and       m  =  0. 
Or,  if 

ma  +  nfi  =  m'a  -\-  n'(3, 
then 

m  =  m'     and     n  =  n'. 

And,  in  general,  if 

2a +  2^  =  0, 

then  I    ....     {H). 

2a  =  0     and     2^  =  0 


QUATERNIONS. 

Three  or  more  A-ectors  infi}-,  however,  neutralize  each  other. 
Thus    (Fig.  7) 

/^       and    this   whether  abcd   be   plane   or 
gauche.     In  any  closed  figure;  there- 
fore, we  have 
A  "a  B 

a  +  y3  +  y  +  s  + =0, 

where  a,  /?,  y,  8,  ,  are  the  vector  sides  in  order. 


11.   Examples. 

1.    Tlie  right  lines  joining  the  extremities  of  equal  and  parallel 
right  lines  are  equal  and x>arallel \ 

Fig.  8. 

o ^  Let  OA  and  r.D   (Fig.  8)   be 

y/ \         "^^-^'"/^  ^^^^    given    lines,    and    oa  =  a, 

/^i^   ^,.--\j^^^  /''  ^^  —  ^1    DA  =  y.     Then,    by 

/\^-^^          \     /  condition,  bd  =  a. 

"^ n  Now, 


also, 


BA  =  BO  4-  OA  =  ^  +  a  ; 
BA  =  BD  +  DA  =  a  +  y  ; 

or,  equating  the  values  of  ba, 

|8-|-a  =  a  +  y. 

Hence  (Art.  2),  y  =  ^,  and  bo  is  parallel  and  equal  to  da. 


2.    The  diagonals  of  a  jjaraUelogram  bisect  each  other. 
In  Fig.  8  we  have 

BD  =  OA  =  OP  +  PA  ; 


also 


BD  =  BP  -t-  PD  ; 
.  • .  OP  +  PA  =  BP  +  PD. 

But,  OP  and  pd  beino;  in  the  same  ri"ht  line, 


Similarly 


OP  =  WiPD. 
PA  =  WBP. 


GEOMETllIC   ADDITION   AND   SUBTE ACTION. 


Hence 
and 


?«PD  +  »i3P  =  PD  +  np, 
7ft  =  1 ,     n  =  1 , 

OP  =  ri),       BP  =  PA. 


3.  If  two  trianglcfi,  having  two  sides  of  the  one  pro2wrtio)ial 
to  two  sides  of  the  other,  be  joined  at  one  angle  so  as  to  have 
their  homologous  sides  parallel,  the  remaining  sides  shall  be  in  a 
straight  line. 


Let  (Fig.  9)  AB  =  a,  AE  =  (3.    Then, 
b}'  condition,  dc  =  xa,  vn  =  xf3. 
Now 


Fig.  9. 


CB  =  CV  +  V\i  =  X   (ft  —  a)  , 


But 


BE  =  ;8  —  a. 


Hence  (Art.  2) ,  b  being  a  common  point,  cb  and  r,E  are  one 
and  the  same  right  line. 


4.  If  two  right  lines  join  the  alternate  extremities  of  two 
parallels,  the  line  joining  their  centers  is  half  the  difference  of 
the  jparaUels. 

We  have  (Fig.  10) 

AB  =  AD  +  DC  +  CB, 

and,  also, 

AB  =  AE  +  EP  +  FB. 

Adding 

2  AB  =  (ad  +  ae)  -I-  (or  +  v.v)  +  (cb  +  fb) 

=  EF  —  CD  ; 

or,  as  lines, 

AB  =  ^  (eF  —  CD)  . 


10 


QUATERNIONS. 


5.    The  medials  of  a  triangle  meet  in  a  point  and  trisect  pacli 
other. 
J'^"-  Let  (Fig.   11)  BO  =  a,  CD  =  /3.     Then 

OC  =  a,  DA  =  (i. 

Now 

BA=2a+2/3=2(a  +  ^), 

and,  since  od=  (a  +  /3),  ba  and  od  are 
parallel. 
Again 

BP  +  PA  =  BA  =  2  OD  =  2  (op  +  Pd)  . 

But  BP  and  pd,  as  also  op  and  pa,  lie  in  the  same  direction, 

and  therefore  ^  ,  „ 

BP  =  2  PD     and     pa  =  2  op. 

Hence  the  medials  oa  and  db  trisect  each  other. 
Draw  cp  and  pe.    Then 


and 


BP=  2pd  =  f  BD  =  f  (2a  +  /?), 

CP  =  CB  +  BP  =  f  (2  a  +  /?)  -  2  a  =  f  (^  -  a)  , 
PE  =  PB  +  BE  =  a  +  ;8  -  f  (2  a  +  y8)  =  i  (;8  -  a)  . 


Hence  pe  and  cp  are  in  the  same  straight  line,  or  the  medials 
meet  in  a  point. 

6.    In  any  quadrilateral,  jjlane    or  gauche,  the  bisectors  of 
opposite  sides  bisect  each  other. 

We  will  first  find  a  value  for  op  (Fig.  12)  under  the  supposi- 
tion that  p  is  the  middle  point  of 
GE.  We  shall  then  find  a  value  for 
OP,  under  the  supposition  that  p  is 
the  middle_  point  of  Fii.  If  these 
expressions  prove  to  be  identical, 
these  middle  points  must  coincide. 
In  this,  as  in  man}-  other  problems, 
the  solution  depends  upon  reaching 
the  same  point  by  different  routes  and  comparing  the  results. 


GEOMETRIC    ADDITION    AND    SUBTKACTION.  11 

Lot  OA  =  a,  OB  =  /?,  OC  =  y. 

1st.  OC  +  CG  =  OE  +  EG.  (/') 

But 

which,  ill  (a),  gives 

y  +  HI3-y)  =ia  +  KG. 

.-.   El>  =  iEG  =  J-(y  +  /3-a), 
OP  =  OE  +  EP  =  ^'  a  +  :^  (y  +  ^  —  a) 

=  i(«  +  i8  +  y).  (&) 

2d.  FII  —  ^AB  =  KO +  OA, 

or 

FH  —  ^  (/3  —  a)  =  —  4-y  -f-  a. 
.  • .    FP  =  -^  FH  =  1  (a  4-  /3  —  y)  , 
OP  =  OF  +  FP  =  ^y  +  i  (a  +  /3  —  y) 

^iC'^  +  Z^  +  y), 

which  is  identical  with   (b).     Hence,  the  middle  points  of  fh 
and  GE  coincide. 


7.  If  ABCD  (Fig.  13)  be  any  parallelogram^  and  op  ayuj  line 
IKirallel  to  dc,  and  the  indicated  lines  be  draivn,  then  will  mn 
be  parallel  to  ad. 


Fig.  13. 

Let  A  M  =  a, 

BM 

:  ra 

=P-                    .      A      o 

Then 

AX/  \^ 

AO  = 

/V\   /  ^V/ 

AD  = 

-.  Sa 

+  7/^,                  /^\  /-^  \/ 

OD  = 

ra  +  .s-a  +  q^.                   '               }f 

We  have 

NM  =  X  O  +  OM  =  NP  4-  I'Mi 

in  which 

NO  =  .r  (  -  ra  +  sa  H-  ry/3)  , 
OM=(l  —  r)  a. 

KP  =7/  (_r^  +  .sa4-7/3), 
PM  =  (1  -  r)  ^. 

12  QUATERNIONS. 

Substituting  in  the  above  equation,  we  obtain,  by  Art.  10, 

Fig.  13.  -  x''  +  ^'s  +  1  —  r  =  ?/s, 

M  ^  xq=-  yr  +  yq  +  \-  r. 

Eliminating  y 

^1-r 
r 

Substituting  this  vakie  in 


Njr  =  NO  4-  OM 
1  -r 


1  ~r 


{- ra  +  SOL  +  qj3)  +  (1-r)  a 
(sa  +  g/3) 


Hence  ad  and  nm  are  parallel. 


8.  /f,  through  any  point  in  a  j^arallelogi-am,  lines  be  draivn 
parallel  to  the  sides,  the  diagonals  of  tico  of  the  ^mrallelograms 
so  formed  ivill  intersect  on  the  diagonal  of  the  original  parallelo- 
gram. 

Fig.  14.  Let    (Fig.    14)    OA  =  a,    OB  =  /3. 

A       D  .c      Then  or  =  ma,  oe  =  nji. 

"We  have 

ED  =  K04-0E+ED  =  ??^-f-(l— 7?l)  a, 
ES  =EO  +  OK  +  RS  =ma+(l— ?;)  (i. 


and 


Also 
FO  =  FR  +  RO  =  XKD  +  RO  =  X  {_nfi  +  (1  —  m)  a]  —  ma,    (a) 


FO  =  FE  +  EO  =  y-ES  +  EO  =  2/  [wa  +  (1  —  w)  /3]  —  w/3.      (6) 
From  (a)  and  (b) 

nx  —  y  (1  —n)  —  n     and     x  (1  —  m)  —  ??i  =  ?/m. 
Eliminating  y 

X  =  . 

1  —  m  —  >i 


GEOMETRIC    ADDITIOX    AND    SUBTltACTION.  13 

Substituting  tliis  value  of  x  in  (a) 

FO  = — ["/^  +  (  1  -  '")  tt]  -  »»a 

(/?  +  «), 


1  —  VI,  —  n 


\  —  III  —  n 
or.  vo  and  oc  =  (/?  +  a)  arc  in  the  same  straiglit  line 


9.  //'.  in  anil  triangle  o.w,  (Fig.  tf)),  a  line  od  be  drawn  to 
the  middle  j)oiut  o/'au,  and  he  produced  to  any  point,  as  f,  and 
the  sides  of  the  triangle  be  produced  to  meet  af  and  bf  in  ii  and 
E,  then  tcill  nil  be  parallel  to  aij. 

Let  OA  =  a,  ou  =  /?.     Tlien  or  =  xa, 
OH  =  ?//3,  A\^  =  p  —  a.  ^'--  ^^• 

Now 


OD  =  OA  +  tV  AB  =  ^  (a  +  /S)  . 


Also,  OF  =  z  (a  +  /3) ,  that  is,  some 
multiple  of  od. 
Then,  1st. 

BR  =  2>BF, 
_/3  +  .ra  =  i*(-/?  +  OF) 

=  p\:-P-tz(a  +  (3)]; 
.  • .  x  =  pz     and     —  1  =  pz  —  j>. 


Eliminating  2 
And,  2d. 


^^  =  .^+1^ 


All  =  (/AF, 
-a  +  y(i^q  (-a  +  Ol'O 

=  7  [-a +  2;  (a +  /?)]; 
.-.  y  =  qz     and     —  1  =  (/^  —  (/. 


Eliminating  2; 
From  (a)  and  {b) 


q  =  y  +  ^' 


(«) 


(Z') 


p      q 


14  QUATERNIONS 

and,  since  ^J  =  a;  +  1   and  q  =  y  -\-  1, 

Fig.  15. 


x  =  y     and    2'>  =  Q- 

"     .  • .  KH  =  RO  +  OH  =  ?//?  —  Xa  =  X  (/3  —  a) 
=  XAB, 


or,  Kii  and  ab  are  parallel. 


10.  If  any  line  pr  (Fig.  IG)  he  clmicn,  cutting  the  ttvo  sides 
of  any  triangle  abc,  and  he  x>roduced  to  meet  the  third  side  in  q, 
then 

^*S-  16.  PC  .  BQ  .  RA  =  CR  .  AQ  .  BP. 

Let  BP  =  a,  CR  =  y8.  Then  pc  =pa, 
RA  =  r/3  and  ba  =  bc  +  ca  =  (1  +p)  a 
+  (l  +  r)^. 

We  have 

AQ  =  a-BA  =  X  [(1  +^))a  +  (l  +  r)  ;8], 

as  also 

AQ  =  AR  +  RQ  =  —  r/3  +  ?/PR  =  —  r/3  4-  y  {pa-  +  /S) . 
.-.  X  (1  -f  j>)  =  yp     and     £«  (1  +  r)  ==  -  r  +  y. 


Eliminating  y 
whence 


a*  =  {\  -\-  x)  pr  ; 

AQ_BQ   PC   RA 
BA   BA   BP   CR 

PC  .  BQ  .  RA  =  CR  .  AQ  .  BP. 


11.   If  triangles  are   equiangular,   the  sides  about  the  equal 
angles  are  proportional. 

Let  (Fig.  17)   BC  =  a,   CA  =  ^.      Then   be  =  ma,    ed  =  »/?, 
BD  =  ma  +  ?i/3  and  ba  =  a  +  j3. 
Now 


Whence 


BD  =  pBA, 

ma  +  «/?  =  ;^  (a  +  /3) . 

?H  —  p,     n  =  p     and     wi  =  n. 
.'.  BE  :  BC  :  :  ED  :  ca. 


GF 
CB 

_  c.\  _ 

"  CA  "" 

n, 

EB 
CB 

ED 

"ca"" 

m, 

DB 
AB 

DE 

AC 

:m, 

12.    If,  throngh  conj  point  o  (Fig.  17),  within  a  trianr/le  aisc, 
lines  be  draicn  jMraJlel  to  the  sides,  then  loill 

CA  CB  AB 

Let  CA  =  yS,   CB  =  a.     Then  ab  = 
a  —  /?,  ED  =  mp,  111  —  p  (a  —  P)  and 

GF  =  ?ta. 

We  have 

CO  =  CG  4-  GO  =  cii  +  no.  (a) 

Now,  as  lines, 

.*.     CG  =  CA  —  GA=  (1— n)  /?. 

.',    GO  =  CE  =  CB  —  EB  =  (1  —  m)  a. 

.-.    HO  =  AD  =  AB  —  DB  =  (1  —  m)  (a  —  /3). 

Substituting  in  (a) 

(1  -  h)  /3  +  (1  -  m)  a  =  pp+{\-  m)  (a  -  /3) , 
or  (Art.  10)  n  +  m  +  p=-2. 

12.  Complanar  vectors  are  those  ivhich  lie  in,  or  picirallel  to, 
the  same  plane.  If  a,  ^,  y  are  anj-  vectors  in  space,  the}'  are 
complanar  when  equal  vectors,  drawn  from  a  common  origin, 
lie  in  the  same  plane. 

If  a,  (3,  y  are  complanar,  but  not  parallel,  a  triangle  can  al- 
ways be  constructed,  having  its  sides  parallel  to  and  some  mul- 
tiple of  a,  ft,  y,  as  aa,  bf3,  cy.  If  we  go  round  the  sides  of  the 
triangle  in  order,  we  have 

aa  -\-b(3  +  cy  =  0. 

If  a,  ft,  y  are  not  complanar,  conceive  a  plane  pai'allel  to 
two  of  them,  as  a  and  ft.  In  this  plane  two  lines  may  be  drawn 
parallel  to  and  some  multiple  of  a  and  ft,  as  aa  and  bft ;  and 
these  two  vectors  may  be  represented  by  pB  (Art.  3). 


16  QUATERNIONS. 

Now  j)S,  being  in  the  same  plane  with  aa  and  &/?,  cannot 
therefore  be  equal  to  y,  or  to  an}-  multiple  of  it ;  p8  and  y  can- 
not therefore  (Art.  10)  neutralize  each  other.     Hence 

j>S  4-  cy  =  aa  +  &/3  +  cy        cannot  be  zero. 

If,  then,  %ve  have  the  relation 

aa  +  b(3  +  cy  =  0 

beticeen  non-imraUel  vectors,  they  are  com2)lanar ;  or,  if  a,  (3,  y 
be  not  complanar,  and  tlie  above  relation  be  true,  then,  also, 

a  =  0,       b  =  0,       c  =  0. 

13.  Co-initial  vectors  are  those  ivhich  denote  transference 
from  the  same  point. 

(a).  If  three  co-initial  vectors  are  comj)lanar,  and  give  the 
relations, 

00       fla+6/3  +  Oy  =  0| 

{b)     a  +  b  +  c  =  Q        j ^  ^' 

'they  ivilJ  terminate  in  a  straight  line. 

For,  let  OA  =  a  (Fig.  15) .  ob  =  yS,  od  =  y.     Then  da  =  a  —  y, 

BA  =  a-/3. 

From  Equation  (9),  (5) 

(«  +  b  +  c)a  =  Q, 
from  which,  subtracting  (a)  of  Equation  (9), 

&  (a  -  /?)  +  C  (a  -  y)  =  0, 
&BA  +  CDA  =  0  ; 

and,  since  these  two  vectors  neutralize  each  other,  and  have  a 
common  point,  the}'  are  on  the  same  straight  line.  Hence, 
A,  D  and  B  are  in  the  same  straight  line. 

(6) .  Conversely,  if  a,  /3,  y  are  co-initial,  complanar  and  ter- 
minate in  the  same  straight  line,  and  a,  b,  c  have  such  values 
as  to  render  aa -\- b/3 -h  cy  =.  0, 

thenwiU  a-hb  +  c^O. 

^^  DA  =  a  —  y      and      BA  =  a  —  ^. 


GEOiSrETRIC   ADDITION   AND   SUBTRACTION. 

But,  b}'  eoiidition, 

a  -  /3  =  a-  (a  -  y) , 
or 

(1  -  .r)  a -13  +  ry  =  0, 
in  which 

(1-x)  -  1  +  .r  =-  0. 


17 


14.   Examples. 

1.  The  extremities  of  the  adjacent  sides  of  a  paraUelogj-ani 
and  the  middle  jwint  of  the  diagonal  between  them  lie  in  the  same 
straight  line. 

^  Fig.  IS. 

Let    OA  =   a,    OB   =   (3,    oc 
Then 

OD  =  OB  +  BD, 
•2v  —  /?  —  a  =  0.  O^ 'B 


But,  also, 


2-1-1 


0, 


hence,  b,  c  and  a  are  in  the  same  straight  line  (Art.  13). 

2.  If  two  triangles,  abc  and  smn  (Fig.  ID),  are  so  situated 
that  lines  joining  corresponding  angles  meet  in  a  jioint,  a?  o, 
then  the  2)ciirs  of  corresponding  sides  j)roduced  tvill  meet  in  three 
pioints,  p,  Q,  R,  which  lie  in  the  same  straight  line. 

Let    OA  =  a,    OB  =  (3,    oc  =  y.  "'    " 

Then     os  =  ma,  om  =  n^, 

ON  =  py,  BA  =  a  —  (3, 

MS  =  via  —  n(3, 

BR  =  X  (a  —  /?)  and 

HIR  =  y  {ma  —  nl3)  . 


1st.         BM  =  BR  —  MR, 


=  X  {a  —  (3)  —  y  (ma  —  nf3), 
-  1  =  —  X  +  yn,  X  —  my  =  0. 


Eliminating  y 


m.  (n  —  1) 


18  QUATERNIONS. 

Also 

OR  =  OB  +  BR  =  ^  +  O;    (a  -  /?)    =  /3  -  '"'S'S^     (a  -  ft), 


whence 


n  (m  —  1)  B  ~  7)1  ( n  —  1)  a  ,   . 

— ^^ -Jl 1 i —  (a) 


2d.  CN  =  CP  —  NP, 

or 

Py  -y=^v  {(3-  y)-tv  (n(3  -  jjy) . 
.-.    p  —  1  =  —  V  +  ivj),         V  —  zvn  =  0. 

Eliminating  iv  ^^  ( „  _  1  \ 

Also  "" "  "      "-^' 

op^oc  +  cp  =  y  +  ^(/3-y)  =  y-'\^/'_~.^^  (/3-y), 

whence  ,         ^ .  ,         , ,   ^ 

n  —  2'> 

3d.    In  the  same  manner,  we  obtain 

mO.-l)a-pO;.-l)y 

2)  —  m 

From  (a),  (5)  and  (c)  we  observe  that,  cleaving  of  fractions, 
and  mnltiplying  (a)  by  p  —  I,  {h)  by  m  —  1,  (c)  by  n  —  1,  and 
adding  the  three  resulting  equations,  member  b}'  member,  the 
collected  coefficients  of  a,  ^8,  y,  in  the  second  member  of  the 
final  equation,  are  separately  equal  to  zero.  Hence  the  first 
member 

OR  {m-n)  {p  —  l)-\.  OP  {n-p)  (m  -1)  +  oq  {p-m)  (n  -1 )  =  0. 

But 
(m  -  v)  (p  _  1)  +  (n  -p)  (m  -  1)  +  {p  -  m)  {n  -  1)  =  0. 

Hence,  r,  p  and  q  are  in  the  same  straight  line. 


/      "■    or  D'f- 
i    UNIVERSITY 

GEOMETRIC    ADDITION    AND    SUBTRACTIOX.  19 

3.    Given  the  relation 

«a  +  b(3  +  cy  =  0. 

Then  a,  /3,  y  are  eomplanar ;  but,  if  co-initial  (as  they  may 
be  made  to  be,  since  a  vector  is  not  changed  by  motion  pnralUl 
to   itself,  i.e.  by  translation 

Fi".  20. 

without  rotation) ,  and  a  +  °'     ' 

6  +  c  is  not  zero,  they  do 
not  terminate  in  a  straight 
line.  Hence,  if  o  is  the  ori- 
gin, and  A,  B,  c,  their  ter- 
minal points,  A,  B  and  c 
are  not  coUinear.  Let  these 
points  be  joined,  forming 
the  triangle  abc  (Fig.  20), 
and  OA,  OB,  oc  prolonged  to 

meet  the  sides  in  a',  W,  c'.     To  find  the  relation  between  the 
seo;mcnts  of  the  sides,  let 


whence 


■.xa,       ob'=;8'=?//S,       oc'=y'=zy, 


y 


._y 


Substituting  these  in  succession  in  the  given  relation, 


-a'+?>iS  +  Cy  =  0, 
a; 

aa  +  ^^(3'+  cy  =  0, 
aa  +  h^-\-%'=  0, 


whence,  since  a,'  c,  v.  arc  to  be  collinoar. 


X 


20 

QUATERNIONS. 

and,  for  a  like  reason, 

a  +  ^  +  c  =  0, 

y 

a  +  b  +  -  =  0. 

Whence 

b  +  c 

^-:^,  ■ 

~      a  +  b' 

and 

,              a 
a  =  -— — -a, 
0  -j-  c 

^'=-^/' 

"-      «  +  6^' 

or,  from  the  given  relation, 

,_  b(i  +  Cy  ,_  Cy  +  aa  ,_  Cla  +  b/3 

''~    b  +  c  '         ^~    c  +  a  '         ■^~    a  +  6  • 
Whence 

b   (a'-/3)   =C   (y   -a'), 

c  (/3'-y)  =  a(a  -^'), 

a  (/-a)   =6(^-y'), 

and 


ba'      c 
a'c      6 

cb'      a 
b'a      c' 

Ae'_& 
c'b       a 

or,  multiplying, 

ba'  .  cb'  . 

Ac'  .  =  a'c 

.  b'a   .   C'l 

4,  If  o  (Fig.  20)  &e  'any  point,  and  abc  any  triangle,  the 
transversals  through  o  and  the  vertices  divide  the  sides  into  seg- 
ments having  the  relation 

ba'  .  cb'  .  ac'  .  =  a'c  .  b'a  .  c'b. 

Let  a'c  =  a,  bc  =  aa,  cb'=  /8,  ca  =  &/3.     Then  ba  =  aa  +  bfi. 

Also  let 

BO  =  iCBB^  OA  =  2/a'a,  BC'=  7nBA,  Cc'=  ^CO. 


GEOMETRIC    ADDITION    AND    SUBTRACTION. 


Then 

BO  =  .rui;'  =  .»■  (liC  +  cb')  =  X  (aa  +(3) , 

OA  =  >i\'\  =  ,(/  (  a'c  +  ca)  =  ?/  (a  +  bfS) , 

Bo'=  //njA  =  III  (aa  +  6/3) , 

cc'  =  zco    =  ;2  (cH  +  bo)  =  2;  [-  aa  +  a;  (aa  +  ^)  ] . 

From  the  triangle  boa  we  have 

BO  +  OA  +  AB  =  0, 

X  {aa  +  l3)  +  y{a  +  bl3)-b(3-  aa,  =  0. 
.-.     xa  +  y  —  a  =  0,         x-{-yb  —  h  =  0. 


Elimhiating  y 

From  the  triangle  bcc' 


b{l-a) 
1-ba  ' 


BC  +  cc'+  c'b  =  0, 
aa  +  z[—  aa  +  X  (aa  +  /?)]-  m  (aa  +  b(3)  =  0, 

whence,  as  nsual,  and  substituting  the  above  value  of  x, 


1  —  m  z=z  —  z 


b{l-a) 
1-ba  '' 


l-g 
1-ba 


1-7)1       1-6 


Substituting  m,  b  and  a, 


c  A 
Bc' 


AB 

b'c 


CA" 

a'b' 


which  is  the  required  relation. 

5.  If  (Fig.  20)  lines  be 
clraicn  throvgh  a,'  b'  c',  and 
jiroducecl  to  meet  the  ojjjwsite 
sides  of  the  triangle  in  p,  Q, 
R,  the7i  are  p,  Q  ourf  r  col- 
linear. 


22  QUATERNIONS. 

With  the  notation  of  the  last  example, 

BC'  =  WiBA  = (aa  +  &/3) . 

1st.    From  the  trians-le  c'ba' 


c'a'=  c'b  +  ba' 
a-1 


a  +  b  —  2 
a-1 


-{aa  +  b/3)  +  {a-l)a 


_[(&_2)a-6/3]. 


Also 


a  +  6  -  2 
a'r  =  a;c'A'=  a'c  +  cr  =  a'c  —  y/3, 

0  —  2 
^^^  BR  =  BC  +  cR  =  aa ^/3.  (a) 

2cl.    From  the  triangle  c'ab' 

c'b'=  c'a  +  ab' 

=  (l-m){aa  +  b/3)  +  il-h)l3 


^-1      [aa_(„_2)/3]. 


Also 


a  +  b-2 


b'q  =  a'c'i5'=  b'c  +  cq  =  b'c  +  ya, 


and 
3cl. 


a  +  b  —  2 

a 
a  — 2 

BQ  =  BC  +  CQ  =  (a  +  ?/)  a  =  "  ^^'  ~        a.  (6) 


a'p  =  .a*A'B'=  a.'  (tt  +  /3) , 
a'p  =  a'b  +  i;p  =  ( 1  —  o)  a-hy  {aa  +  b^) . 
__  a  - 1 
'**  '^~a-b' 


GEOMETRIC    ADDITION   AND    SUBTRACTION. 


23 


ind 


BP  =  ?/IiA  = (aa  +  fj(3)  . 

a  —  b 


(0 


Multiplying  the  second  members  of  (a),  {b),  (r),  l\v  (a  — 1) 
(^)  _  2),  -  (a  —  2)  (6  -  1),  {a  —  b)  respectively,  their  sum  is 
zero.     Hence 

(a  _  1)  (6  -  2)  BR  -  («  -2){b- 1)  liQ  +  (a  -  b)  bp  =  0. 

But 

(a  -1)  {b  -  2)  -  (a  _  2)  (6  -1)  +  («  -  &)  =  0. 

Hence  k,  q  and  p  are  collinear. 


6.    If  vc  (Fig.  20)  and  po  be  produced  to  meet  a  a'  and  bc, 

?/<e?i  T  and  s  are  collinear  with  c!  A  similar  proposition  would 
obtain  for  q  and  k. 

AVith  the  following  notation, 


we  have 


A  =  a,  ba'=  /3,  bb'=  aa  +  &/3, 

BO  =  BA  +  Ab'+  b'o  =  r.A'+  a'o, 

a  +  bli-{\-a)a  +  x{aa  +  b(i)  =  (i  +  y{a- fS). 

_     a 

?>/3  +  aa 
a  +  0 


also 


Fig.  20. 


BP  =  BA'+  a'p  =  BA  +  AP  ; 
/3  +  ^[oa+(6-l)/3]=a+lt'a, 

a  -  1  +  b 

.-,    w=  ■ , 

\-b 


and 


1-6 


EC  =  BA'+  A'C  ='BA  +  AC, 


24 


QUATERNIONS. 


BC  = 


I- a 


Now  to  find  BS,  Bc'  and  bt,  we  have 


1st. 


Fig. 


X  BA  =  BP  +  ypo, 


w 

1-26-a' 

^/-^^ 

— ==/q 

..-_        &/? 

.^^^^^ 

1-26-a 
2d. 

BC'  =  v'ba  =  bc  +  i(CO, 

s\a'V^^/ \ 

1 

\\/\\v^ 

J 

•     -r'-          ^* 

T              ^^ 

2o  +  6-l' 

^^R 

PC'-           ''* 

2a +  6-1 

i.                  BT  =  BA'+  a't  = 

=  Ba'+  2'a'o  =  BP  +  IC'PC, 

.• 

a 

+  & 

R' 

,_¥- 

-  aa 

3d 


6  —  a 
Clearing  of  fractions  and  adding 


(1  -  26  -  a)  BS  +  (2a  +  6  -1)  bc'+  (6 -  a)  bt  =  0, 

as  also 

(1  -  2  6  -  a)  +  (2  a  +  6  - 1)  +  (6  -  a)  =  0. 

Hence  s,  c'  and  t  are  collinear. 


15.  A  medial  vector  is  one  drawn  from  tlie  origin  of  two  co- 
initial  vectors  to  the  middle  point  of  the  line  joining  their 
extremities. 


GEOMETKIC    ADDITION    AND    SUBTIIACTION. 


Thus  (Fig.  21),  if  v  is  the  middle  point  of  ab,  op  is  a  medial 
vector.     To  find  an  expression  for  it,  let  oa  =  a,  on  =  /3,  then 


or,  adding, 


op  =  OA  +  Al'  =  a  +  Al', 
op  =  Oli  +  151*  =/3  —  AP, 


(10) 


The  signs  in  this  expression  will,  of  course,  depend  u[)on  the 
original  assumptions.     Thus,  if  ao  =  a, 

OP  =  —  a  +  AP  =  y3  —  AP, 
OP  =- 


Fig.  21. 


16.   An  Angle-Bisector   is  a  line  which  Ijisects  an  angle. 

To  find  an  expression  for  an  angle-bi- 
sector as  a  vector,  let  oe  =  a  (Fig.  21) 
and  OF  =  ^  be  unit  vectors  along  oa  and 
oii.  Complete  the  rhombus  okdf.  Since 
the  diagonal  of  a  rhombus  l)isects  the 
angle,-  od  is  a  multiple  of  op.  Now  od 
=  a  -H  /3,  hence 


op  =  x{a+^) 


(11). 


In  this  expression  op  is  of  any  length  and  x  is  indeterminate. 
If  op  is  limited,  as  by  the  line  ap.,  then 


AP  =  x(a  -f-  /3)  —  aa, 
AP  =  ?/AB  =  y{h(i  —  aa) , 
x{a  +  /?)  —  aa  =  j/{b(S  —  aa) . 


(«) 


Eliminating  x 
Substituting  in  (a) 


a  =  —  ya     and     x  =  ;/b. 


a  +  b 


AP  —  AB 

a  +  b 


(12). 


26 


QUATEENIOlNlS. 


17.   Examples. 

1.  If  parallelograms^  whose  sides  are  parallel  to  tico  given 
lines,  be  described  upon  each  of  the  sides  of  a  triangle  as  diago- 
nals, the  other  diagonals  idll  intersect  in  a  p)oint. 


Fi-.  22. 


\J 

X 

^^~"~~~-^^ 

Let  ABC  (Fig.  22)  be  the  given  tri- 
angle. Let  the  diagonals  b'f  and  a'd 
intersect  in  p,  and  suppose  oe  to  meet 
a'd  in  some  point  as  p! 

Let  oA  =  a,  ob'= /3,  whence  oa'  = 
7«a,  OB  =  njS. 

Now 

b'p  —  DP  =  a. 

Bnt 


?/B'Q: 


And 


y  >|  (b'c  +  b'b) 

J-,V  [ma +  (»-!)/?]. 


(a) 
(Art.  15) 


D  P=  2DH  =  Z  ,^  (dC  +  Ca') 

=  i:/[(m-l)a-/3]. 


Substitnting  in  (a),  we  obtain,  as  usual, 
._   2(l-n) 


1  +  m)L  —  n 

Again 

op'—  dp'=  a  +  (i 

But 

C^) 


Op'=  a.'OG  =  X  ,  4-  (oA  +  ob) 

=  i.i:  (a  +  nl^)  . 

.Sul)stituting  in  (l))  this  value  of  op'  and  dp'=  vdh,  we  obtain 

as  before, 

v= ^^ —• 

1  +  m  n  —  n 

Or,  vim  =  ZT>u  =  dp'=  dp.     Hence,  p  and  p'   coincide,   and 
the  three  diagonals  meet  in  a  point. 

2.    A  triangle  can  ahoays  be  constructed  ^vhose  sides  are  equal 
and  parallel  to  the  medials  of  any  triangle. 


f;KO:\IETRIC    ADDITION   AND    SUBTRACTION. 


In  Ki^-.  'i^  wo  have 

aa'=  ab  +  ha'=  ah  +  ^BC. 
bb'=  bc  +  tVca. 

CC'=  CA  +  jAB. 
.-.    AA'+BB'+Cr'==§(AB+  BC  +  Ca)=0.     (Alt. 

3.    The  angle-bisectors  of  a  triangle  meet  in  a  point. 

Let  a,  /3,  7  be  unit  vectors  along  bc,  pjg  03. 

AC.  AB  (Fig.  23).  c 

Tlien  (Art.  IG) 

AP  =  .r(y  +  ^), 

BP  =  ^(a-y).  (a) 


10), 


Now 


BC  =  AC  —  AB, 

Ott  —  hfi  —  c'y 


('->) 


where  a,  Z>,  c  are  the  lengths  of  the  sides. 
Substituting  a  from  (b)  in  («) 


We  have  also 


=  Al'-AC  =  .T(y  +  /3)-6/?, 


^^^/5  -  Cy  . 

CP  =  BP  +  CB  =  II    [  ;; y  )  +  Cy  —  0(3. 


x-h  =  - b. 

a 


Eliminating  ?/ 
fSubstituting'in  (c) 


«  +  /y  + 


CP  = 


a  +  6  -+■ 

h 
a  +  b  + 

h 


(y  +  l3)-b(S 
[ry-(a  +  h)f3^ 
(_aa-o/3) 


a  +  b  -\-c 

=  p(a  +  (3). 

Hence  (Art.  IG)  cp  is  an  angle-bisector. 


(0) 


28 


QUATERXIONS. 


18.  The  Mean  Point  of  any  j^ohjgon  is  that  to  which  the 
vector  is  the  mean  of  the  vectors  to  the  angles. 

Hence,  to  find  the  mean  point,  add  the  vectors  to  the  angles 
and  divide  by  the  number  of  the  angles.  Thus,  if  O],  o.o,  03  .... 
be  the  vectors  to  the  angles,  the  vector  to  the  mean  point  is 


«!  +  "2  +  0-3+  ••••  -\-0-n 


(13), 


where  n  is  the  number  of  the  angl?:^. 

The  mean  point  of  a  pol3edron  is  similarly  defined.  It  co- 
incides in  either  case,  as  will  appear  later,  with  the  center  of 
gravit}'  of  a  S3'stem  of  equal  particles  situated  at  the  vertices 
of  the  polygon  or  polyedron. 


19.   Examples. 

1.    The  mean  point  of  a  tetraedron  is  the  mea.n  point  of  the 
tetraedron  formed  by  joining  the  mean  jooiiits  of  the  faces. 

Let  (Fig.  -24)  oa  =  a,  on  =  /?,  oc  = 
y.  The  vectors  from  o  to  the  mean 
points  of  the  faces  are 

i(a  +  ^  +  y), 

H«  +  y). 
Hy  +  /3), 

and  that  to  the  mean  point  of  the  tetraedron  formed  by  joining 
them  is 


,  p^+|+, ^ „_+_^ ^ „_+, + i+jj = J („ + ^ + 


y)- 


which  is  the  vector  to  the  mean  point  of  oabc. 

The  same  is  true  of  the  tetraedron  formed  by  joining  the  mean 
points  of  the  edges  ab,  nc  and  ca  with  o,  since 


["-* 


/5  +  ^+2H.'i+V|  =  j(„  +  ;3  +  ,), 


GEOMETRIC    ADDITION    AND    SUBTRACTION.  29 

The  above  is,  of  course,  independent  of  the  origin,  and  would 
be  true  were  o  not  taken  at  one  of  the  vertices. 

2.    The  intersection  of  the  bisectors  of  the  sides  of  a  quadri- 
lateral is  the  mean  point. 

Let  (Fig.  25)  oa  =  a,  on  =  /?,  oc  =  y, 
oi)  =  8,  OK  =  p.     Then 

p  =  OE  4-  KK 

=  OE  +  h  ("J''  -  '>^-) 

=  i(8  +  l)F  +  y  +  CE) 

=  i[8  +  l(a-8)  +  y  +  ^(^-y)] 

=  |(«  +  ^  +  y  +  5)- 

If  o  is  at  A,  then  oa  =  a  =  0,  and 

p=|(^  +  y  +  8). 

3.  If  the  sides  {in  order)  of  a  quadrilateral  he  divided  propor- 
tionately, and  a  neio  quadrilateral  formed  by  joining  the  points 
of  division,  then  loill  both  quadrilaterals  have  the  same  mean 
point. 

Let  a,  /S,  y,  S  be  the  vectors  to  the  vertices  of  the  given 
quadrilateral,  from  any  initial  point  o. 

Then,  for  the  vector  to  the  mean  point,  we  have 

i(a  +  /34-y  +  ^^)- 

If  m  be  the  given  ratio,  and  a,  (3',  y',  8'  the  vectors  to  the  ver- 
tices of  the  second  {[uadrilateral.  then 

a'=  a  +  m  {f3-a)  =  {\-  m)  a  +  m(3. 
l3'={\-m)l3+my, 
y'=  (1  —  'ni)y  +  w8, 

8'  =  a  +  (1  -  m)  (8  -  a)  =  8  -  m  (8  -  a)  ; 
whence 

1-  (/3'+  y'+  '^'+  ^')  =  i  (a  +  ^  +  y  +  5). 


30 


QUATERNIONS. 


4.  In  any  quadrilateral^  2)lane  or  gauche,  the  middle  point 
of  the  bisector  of  the  diagonals  is  the  mean  point. 

Let  (Fig.  26)  oa  =  a,  ob  =  /?,  oc  =  y,  os  =  h/.     Theu 

^'-26.  OQ  =  OA  +  AQ  =  i(a+^), 

fiud 

OP  =  OQ+iQS  =  |(a+/S)+|(^OC-OQ) 

5.  If  the  two  opposite  sides  of  a  quadrilateral  be  divided  pro- 
portionately,  and  the  points  of  division  joined,  the  mean  points 
of  the  three  quadrilaterals  ivill  lie  in  the  same  straight  line. 

Let  c',  a'  (Fig.  27)  be  the  jjoints 
of  division,  and  on  tlie  given  ratio. 
Tlien,  if  oa  =  a,  nc  =  y,  oa'=  Hia, 
c'c  =  ony,  AB  =  /S  and  o  is  the  in- 
itial point,  the  vectors  to  the  mean 
points  p,  p'  p"  are 

OP    =l(3a+2/3  +  y). 
OP'  =  i[(vi  +  2)  a  +  2^  +  (2  -  m)y], 
op"  =  l[(m  +  3)  a  +  2  /3+  (1  -  m)  y]  ; 
1  —  -m 


pp' 


.(y-a), 


-  (y  -  a) 


Therefore,  v\  v','  p  are  in  the  same  straiaht  line. 


20.    Exercises. 

1.  Tlie  diagonals  of  a  parallelepiped  liisoet  each  other. 

2.  In  Fig.  58,  show  that  bg  and  on  are  parallel. 

3.  If  the  adjacent  sides  of  a  qnadi'ilateral  be  divided  propor- 
tionately, the  line  joining  the  points  of  division  is  parallel  to  the 
diagonal  joining  their  extremities. 


GEOMETRIC    ADDITION   AND    SITBTRACTION.  31 

4.  Tlie  nu'dial  to  the  l);ise  of  an  isosceles  triauiile  is  an  angle- 
bisector. 

5.  If  the  diagonal  of  a  parallelogram  is  an  angle-bisector,  the 
parallelogram  is  a  rhombns. 

6.  Any  angle-bisector  of  a  triangle  divides  the  opposite  side 
into  segments  proportional  to  the  other  two  sides. 

7.  The  line  joining  the  middle  point  of  the  side  of  an}'  paral- 
lelogram with  one  of  its  opposite  angles,  and  the  diagonal  which 
it  intersects,  trisect  each  other. 

8.  If  the  middle  points  of  the  sides  of  any  (luadrilateral  be 
joined  in  succession,  the  resulting  (igure  will  be  a  i)arallelogram 
Avith  the  same  mean  point. 

9.  The  intersections  of  the  l)isectors  of  the  exterior  angles 
of  any  triangle  with  the  opposite  sides  are  in  the  same  straight 
line. 

10.  If  AB  be  the  common  base  of  two  triangles  whose  vertices 
are  c  and  d,  and  lines  be  drawn  fiom  any  point  ic  of  the  base 
parallel  to  ad  and  ac  intersecting  kd  and  liC  iu  f  and  g,  then  is 
FG  parallel  to  dc. 


CHAPTER    II. 

Multiplication   and   Division  of  Vectors,  or  Geometric   Multipli- 
cation and  Division. 

21.   Elements  of  a   Quaternion. 

Tlie  quodent  of  tivo  vectors  is  called  a  Quaternion. 
^Ye  are   now  to  see  what  is  meant  hy  the   quotient  of  two 
vectors,  and  what  are  its  elements. 

Let  a  and  /?'  (Fig.  28)  be  two  vec- 
'^'  '^'  tors  drawn  from  o  and  o'  respectively 

and  not  lying  in  the  same  plane  ;  and 
let  their  quotient  be  designated  in  the 
usual  way  liv  — . 

\  "Whatever  their  relative  positions,  we 

o''  ^r      J}'      may  alwaj-s  conceive  that  one  of  these 

vectors,  as  (3',  may  be  moved  parallel 
to  itself  so  that  the  point  o'  shall  move  over  the  line  o'q  to  o. 
The  vectors  will  then  lie  in  the  same  plane.  Since  neither  the 
length  or  direction  of  /?'  has  been  changed  during  this  parallel 
motion,  we  have  (3  =  [i',  and  the  quotient  of  any  two  vectors,  a, 
P',  will  be  the  same  as  that  of  two  equal  co-initial  vectors,  as  a 
and  jB.     We  are  then  to  determine  the  ratio  -,  in  which  a  and  B 

lie  in  the  same  plane  and  have  a  common  origin  o. 

Whatever  the  nature  of  this  quotient,  we  are  to  regard  it  as 
some  factor  which  operating  on  the  divisor  produces  the  dividend^ 
i.e.  causes  /?  to  coincide  with  a  in  direction  and  length,  so  that 
if  this  quotient  be  g,  we  shall  have,  by  definition, 

fi^  =  a     when     ^  =  q (14). 


GEOMETRIC    .MULTIPLICATION   AND   DIVISION.  33 

If  at  the  point  o'  we  suppose  a  vector  o'c  =  y  to  bo  drawn, 
not  parallel  to  the  plane  aob,  and  that  this  vector  be  moved  as 
before,  so  that  o'  falls  at  o,  the  plane  which,  after  this  motion, 
y  will  determine  with  a,  will  differ  from  the  plane  of  a  and  fi,  so 
that  if  the  quotient 

a 

q  and  q'  will  ditfer  because  their  planes  differ.  Hence  we  con- 
clude that  the  quotients  q  and  q'  cannot  be  the  same  if  a,  y3  and 
y  are  not  parallel  to  one  plane,  and  therefore  that  the  position 
of  the  plane  of  a  and  [i  must  enter  into  our  conception  of  the 
quotient  q. 

Again,  if  y  be  a  vector  o'c,  parallel  to  the  plane  Aon,  but 
differing  as  a  vector  from  (S',  then  when  moved,  as  before,  into 
the  plane  aob,  it  will  make  with  a  an  angle  other  than  boa. 
Hence  the  angle  between  a  and  /?  must  also  enter  into  our  con- 
ception of  q.  This  is  not  only  true  as  regards  the  magnitude  of 
the  angle,  but  also  its  direction.  If,  for  example,  y  have  such  a 
direction  that,  when  moved  into  the  plane  aob,  it  lies  on  the 
other  side  of  a,  so  that  aoc  on  the  left  of  a  is  equal  to  aob,  then 

the  quotient  q',  of  -,  in  operating  on  y  to  pi-oduce  a  must  turn  y 

y 

in  a  direction  opposite  to  that  in  which  7  =  -  turns  /3  to  produce 

a.  Therefore  7  and  7'  will  ditfer  unless  the  angles  between  the 
vector  dividend  and  divisor  are  in  each  the  same,  both  as  regards 
magnitude  and  direction  of  rotation.  Of  the  two  angles  tiu'ough 
which  one  vector  may  be  turned  so  as  to  coincide  with  the  other 
is  meant  the  lesser,  and  it  will  therefore,  generally,  be  <  180? 

Finally,  if  the  lengths  of  B  aild  y  differ,  then  -  =7  will  still 

differ  from  -  =  q'.     Therefore  the  ratio  of  the  lengths  of  the  vec- 

7 
tors  must  also  enter  into  the  conception  of  ^. 

We  have  thus  found  the  quotient  7,  regarded  as  an  operator 
which  changes  3  into  a,  to  depend  upon  the  plane  of  the  vectors, 
the  angle  between  them  and  the  ratio  of  their  lengths.     Since 


34  QUATERNIONS. 

two  angles  are  requisite  to  fix  a  plane,  it  is  evident  that  q 
depends  upon  four  elements,  and  performs  two  distinct  opera- 
tions : 

1st.  A  stretching  (or  shortening)  of  y3,  so  as  to  make  it  of 
the  same  length  as  a  ; 

2d.  A  turning  of  y3,  so  as  to  cause  it  to  coincide  with  a  in 
direction, 

the  order  of  these  two  operations  being  a  matter  of  indiffer- 
ence. 

Of  the  four  elements,  the  turning  operation  depends  upon 
three  ;  two  angles  to  fix  the  plane  of  rotation,  and  one  angle  to 
fix   the    amount    of  rotation    in   that 
Fig.  28.  plane.      The  stretching  operation  de- 

pends 0UI3"  upon  the  remaining  one, 
i.e.,  upon  the  ratio  of  the  vector 
lengths.  As  depending  upon  four 
elements  we  observe  one  reason  for 
\  calliug  q  a  quaternion.     The  two  ope- 

o''  ^t }?'      rations  of  which  q  is  the  symbol  being 

entirel}'  independent  of  each  other,  a 
quaternion  is  a  complex  quantity,  decomposable,  as  will  be 
seen,  into  two  factors,  one  of  wdiich  stretches  or  shortens  the 
vector  divisor  so  that  its  length  shall  equal  that  of  the  vector 
dividend,  and  is  a  signless  number  called  the  Tensor.,  of  the 
quaternion  ;  the  other  turns  the  vector  divisor  so  that  it  shall 
coincide  with  the  vector  dividend,  and  is  therefore  called  the 
Versor  of  the  qvaternion.  These  factors  are  symbolically  repre- 
sented by  Tq  and  Vq.  read  "  tensor  of  q"  and  "versor  of  (/." 
and  q  maj'  be  written 

q  =  Tq  .  Vq. 

22.  An  equality  beticeen  tivo  quaternions  may  be  defined  di- 
rectly from  the  foregoing  considerations. 

If  the  plane  of  a  and  (B  be  moved  parallel  to  itself;  or  if  the 
angle  aob  (Fig.  28),  remaining  constant  in  magnitude  and  esti- 
mated in  tlie  same  direction,  be  rotated  about  an  axis  througli  o 
perpendicular  to  the  plane  ;  or  the  absolute  lengths  of  a  and  ^ 


GEOMETRIC   MULTIPLICATION    AND    DIVISION.  'J/3 

vary  SO  tliat  their  nitio  remains  constant,  7  will  renuiin  the  same. 

Plonceif  ^^  ^^, 

-  —  q     and     -—=7 

then  ^vill 

■ulien 

1st.  2V;e  t-edor  lengths  are  in  the  same  ratio,  and 

2d.  Hie  vectors  are  in  the  same  or  2xiraUf'l  j^^mes,  and 

3d.    The  vectors  make  with  each  other  the  same  o.mjie  both  as 

to  magnitude  and  direction. 

The  plane  of  the  vectors  and  the  angle  between  them  are 

called,  respectively,  the  plane  and  angle  oC  the  (luaternion,  and 

the  expression  -,  a  geometric  fraction  or  quotient.     It  is  to  l)e 

observed  that  q  has  been  regarded  as  the  operator  on  /3,  produc- 
ing a.  This  must  be  constantly  borne  in  mind,  for  it  will  sub- 
sequently appear  that  if  we  write  q/3  =  a  to  express  the  operation 
by  which  7  converts  ft  into  a,  qf3  and  ftq  will  not  in  general  be 
equal. 

23.  Since  7,  in  operating  upon  j3  to  produce  a,  must  not  only 
turn  ft  through  a  definite  angle  but  also  in  a  definite  direction, 
some  convention  defining  positive  and  negative  rotation  with 
reference  to  an  axis  is  necessary. 

By  positive  rotation  with  reference  to  an  axis  is  meant  left- 
handed  rotation  when  the  direction  of  the  axis  is  from  the  plane 
of  rotation  toivards  the  eye  of  a  pex'son  who  stands  on  the  axis 
facing  the  plane  of  rotation. 

[If  the  direction  of  the  axis  is  regarded  as  from  tlie  eye 
towards  the  plane  of  rotation,  positive  rotation  is  righthanded. 
Thus,  in  facing  the  dial  of  a  watch,  the  motion  of  the  hands  is 
positive  rotation  relatively  to  an  axis  from  the  e^e  towards  the 
dial.  For  an  axis  pointing  from  the  dial  to  the  eye,  the  motion 
of  til"  hands  is  negative  rotation.  Or  again,  the  rotation  of  the 
earth  from  west  to  east  is  negative  relative  to  an  axis  from  north 
to  south,  but  positive  relative  to  an  axis  from  soutii  to  nortli.] 

On  the  above  assumption,  if  a  person  stand  on  the  axis,  fac- 
ing the  positive  direction  of  rotation,  the  positive  direction  ot 


36 


QUATERNIONS. 


Fig.  31  ibis). 
J 


the  axis  Avill  always  be  from  the  place  -where  he  stands  towards 

the  left. 

If  /,  A-,  j  (Fig.  31)  be  three  axes  at  right  angles  to  each  other, 
with  directions  as  indicated  in  the 
figure,  then  positive  rotation  is  from  i 
to  J,  from  j  to  k,  and  from  k  to  /,  rela- 
tively to  the  axes  k,  i,  j  respectivel}'. 
A  preciselj'  opposite  assumption  would 
be  equally  proper.  The  above  is  in 
accordance  with  the  usual  method  of 
estimating  positive  angles  in  Trigo- 
nometry and  Mechanics. 

24.  Let  OA  and  on  (Fig.  29)  be  any 
two  co-initial  vectors  whose  lengths  are  a  and  6,  a  and  /3  being 
unit  vectors  along  oa  and  ob,  so  that 

OA  =  ((a, 
OB  =6/3. 

Let  the   angle  aob   lietween   the 
vectors  be  represented   by  cfi-;   also 
draw  AD  perpendicular  to   ob,   and 
let  the  unit  vector  along  da  l>e  8. 
/^  The     tensor     of    od     is    evidently 

acos(ji  and  that  of  da  a  sin.^.  If 
we  assume  that,  as  in  Algebra,  geometrical  quotients  which 
have  a  common  divisor  are  added  and  subtracted  by  adding  and 
subtracting  the  numerators  over  the  common  denominator,  so 
that 


then,  since 
we  have 


a 

-r 

OA 

=  OD 

+  DA, 

oa 

OD  +  DA 

OD 

DA 

OB 

OB 

OB 

a 

COS<^ 

■13,  a 

sin(f) 

b./S 


a  /cos  (f)  .  0       sin  <^  .  8> 


GEOMETKIC    MULTIPLICATION    AND    DIVISION.  o7 

We  hivve  already  delined  (Art.  8)  the  quotient  of  two  parallel 
vectors  as  a  sealai',  and  in  the  first  term  of  the  parenthesis,  /? 

beinii'  a  unit  vector,  -  =  1 ,  and 


OB        b\ 


-fsin<^.|Y  (a) 


Tlie  last  term  contains  tlie  quotient  -  of  two  unit  vectors  at 

right  angles  to  each  other.  This  quotient  is  to  be  regarded,  as 
before,  as  a  factor  which,  operating  on  the  divisor  (3,  produces 
8,  I.e.,  turns  /3  left-handed  through  an  angle  of  90°  ;  and  this 
quotient  must  designate  the  plane  of  rotation  and  the  direction 
of  rotation.  If  we  define  the  effect  of  any  unit  vector,  operating 
as  a  multiplier  upon  another  at  right  angles  to  it,  to  be  the  turn- 
ing of  the  latter  in  a  positive  direction  through  an  angle  of  90° 
in  a  plane  perpendicular  to  the  operator,  then  the  unit  vector  e, 
drawn  from  o  perpendicular  to  the  plane  of  S  and  /?,  and  in  the 
direction  indicated  in  the  figure,  will  be  the  factor  Avhich  oper- 
ating on  /3  produces  8,  and 

€j8  =  8     or     -  =  e. 

The  unit  vector  c,  as  an  axis,  indicates  the  plane  of  rotation  ; 
its  direction  determines  the  direction  of  rotation,  and  by  defini- 
tion its  rotating  effect  extends  through  an  angle  of  90°  ;  as  a 
quotient,  therefore,  it  completely  detei-mines  the  operator  which 
changes  /3  into  8.     E(iuation  ((()  thus  becomes 


^=-'(cosc^+€  sin(^), 

OB        h 

or,  if  OA  and  or.  1h>  themselves  denoted  by  a  and  /?,  and  the  ti-n- 
sors  of  a  and  (3  by  Ta  and  TyS, 

5  =  Tl^  (cos  (/) -f  €  sin  <^)      ....     (15). 


38  QUATEENIONS. 

Tn 

ill  which    —  is  the  tensor  of  o,  beino-  the  ratio  of  the  vector 

T/3  ° 

lengths,  and  cos  ^  +  e  sin  4>  is  the  versor  of  g,  its  plane,  deter- 
mined b}'  the  axis  e,  and  angle  ^  being  the  plane  and  angle  of 
the  quaternion. 

When  a  and  /3  are  of  the  same  length,  or  Ta=T/3,  T(7  =  — =  1, 

and  the  effect  of  g  as  a  factor,  or  operator,  is  simplj-  one  of 
version. 

Like  T,  the  symbol  U  is  one  of  operation,  indicating  the  oper- 
ation of  taking  the  versor,  so  that 

Uq  —  cos  ^  -f  e  sin  </). 

This  operation  takes  into  account  l)nt  one  of  the  two  distinct 
acts  which  we  have  seen  the  quotient  q  must  perform,  as  an 
agent  converting  /?  into  a,  namel}',  the  act  of  version  ;  it  thus 
eliminates  the  quantitative  element  of  lengtli.     In  this  respect  it 
is  similar  to  the  reduction  of  a  vec- 
'"■  "■        •  tor  to  its  unit  of  length,  an  opera- 

tion which  also  eliminates  this  same 
element   of  length,    and   has   been 
designated  by  the  same  symbol  U. 
When  a  and  f3  are  at  right  angles 
/^  to  each  other,  ^  =  90f  and  the  ver- 

sor cos  ^  4-  €  sin  cf>  reduces  to  the 
unit  vector  c,  which  has  been  de- 
fined, as  an  operator,  to  be  a  versor  turning  a  line  at  right 
angles  to  it  through  an  angle  90^"  Any  vector,  therefore,  as  a, 
contains,  in  its  unit  vector  in  the  same  direction,  a  versor 
element  or  factor  of  Avhich  Ua  is  the  symbol,  U  indicating  the 
reduction  of  a  to  its  unit  of  length  or  the  taking  of  its  versor 
factor.     Hence  the  appellation  versor  of  a  (Art.  7). 

If  in  Equation   (1."))   the  vectors  be   reduced  to   the  unit  of 
length, 

V/3  /3 


cKoMirrnic  Mrr/ni'LicATiox  a>I)  division.        39 

25.  ^N'c  may  now  i-xprcss  the  relation 

|  =  ^(cosc/,  +  .siu<^)  =  r/  (Eq.  L")) 

in  the  symbolic  notation 

«  _  ff  ^    I' ^  1 

^         ^'     /^j- (10), 

or  I 

q  =  Tq.  Vq  ) 

and  say  that  the  quotient  of  two  vectors  is  the  product  of  a  tensor 
CDul  a  versor;  and  that 

1st.  The  tensor  of  the  quotient,  (77^ ),  i^  the  ratio  of  their 
tensors;  ^  '^^ 

2d.  The  versor  of  the  quotient,  (eo.S(^4-  €  sin^),  is  the  cosiru^ 
of  the  contai)ied  cinr/le  plus  the  jyroduct  of  its  sine  and  a  unit 
vector^  at  right  angles  to  their  plane  and  such  that  the  rotation 
u-hich  causes  the  divisor  to  coincide  »t  direction  tcilh  the  dividend 
shall  be  positive. 

26.  II".  for  -  =  q,  we  write  -  —  q\  it  is  evident  that  q'  ditiers 

from  q  both  in  the  act  of  tension  and  ver- 
sion ;  the  tensor  of  q'  being  the  reciprocal  '^'' 
of  the  tensor  of  r/,  and  the  unit  vector  e,  /'  y  a 
while  still  parallel  to  its  former  position,  ~V y^ 
is  reversed  in  direction  (Art.  23)   since  /X     ■\ 

the  direction  of  rotation  is  reversed  (Fig.        ^ ' *-^ 

30) .     Hence 

^  =  '^(coscf.-€sin</>)      .     .     .     .     (17). 

^  is  called  the  reciprocal  of  -.     As  already  remarked,  tlu- 
a  /i 


positive  direction  of  c  is  a  matter  of  choice.     It  is  only  neces- 
sary that  if 
conversely. 


sary  that  if  we  have  +  e  in  U^,  we  must  have  —  €  in  UC.  or 


40 


QUATERNIONS. 


Fig.  31. 
J 


27.  Let  /,  J,  Jc  (Fig.  31)  represent  unit  vectors  at  right  angles 
to  each  other.  The  eftect  of  any  unit  vector  acting  as  a  muki- 
pher  upon  another  at  right  angles  to  it, 
has  lieen  defined  (Art.  24)  to  be  the 
turning  of  the  latter  in  a  positive  direc- 
tion in  a  plane  perpendicular  to  the  ope- 
rator or  multiplier  through  an  angle  of 
90?  Thus,  i  operating  on  /  produces  Jc. 
»._  This   operation   is   called  multiplica- 

tion, and  the  result  the  product,  and  is 
expressed  as  usual 


(/=A- (18). 

The  quotient  of  two  A^ectors  being  a  factor  which  converts 
the  divisor  into  the  dividend,  we  have  also 


(10). 


either  the  product  or  quotient  of  tioo  unit  vector's  at  right  angles  to 
each  other  being  a  unit  vector  perpendicular  to  their  plane. 

This  multiplication  is  evidently-  not  that  of  algebra ;  it  is  a 
revolution,  which  foi-  rectangular  vectors  extends  through  90? 
Nor  is  k  in  Equation  (18)  a  numerical  product,  nor  *  in  Equa- 
tion (19)  a  numerical  quotient.  This  kind  of  multiplication  and 
division  is  called  geometric. 

In  accordance  with  the  above  definition  we  ma}'  wa'ite  the  fol- 
lowing equations  : 


i^- 


=  -  A- 


hj  =  -  i 


J 
i 

!=■' 
1=1- 

I 
Ji-  ■ 


(20), 


GEOMETRIC    MULTIPLIC 

ATION    AND    DIVISION 

'A'  =  -./ 

"a- 

'■(-./)=- A- 

-  A'  _  I 

/(-/••)=./ 

^■" 

A'(-o=-; 

—  / 

'  •    •    • 

A- (-./)  =  '• 

-i-  =  k 

j(- /.■)=- ^ 

—  '  —  j 

./(-o=/^- 

A-  _  • 

41 


(20). 


Since  the  effect  of  i,  /.-, ./  as  operators  is  to  turn  a  line  from  one 
direction  into  another  which  differs  from  it  by  90°  they  are 
caUed  (jitadranlal  rersors. 


28.    Since 
we  have 


/  X  ./  =  A'     and     /  x  A'  =  —,/  =  —  1  X  ./, 

/  X  /  X  ./■  =  —  1  X  ./, 

/  X  i"  =  —  1 . 


We  may  denote  th(>  continued  use  of  /  as  an  operator  In"  an 
exponent  which  indicates  tlie  luunber  of  times  it  is  so  used. 
Tliis  is  consistent  witli  the  meaning  of  an  exponent  in  algebraic 
notation.  In  both  cases  it  denotes  the  numl)er  of  times  the 
operator  is  used,  in  one  instance  as  a  numerical  factor,  in  the 
other  as  a  versor.     Thus 


.;,/'"  =.r 


etc. 


././     .r 


In  conformity  to  this  notation  the  above  equation  becomes 

r=.-\    .......     (21), 


42  QUATERNIONS, 

and  in  a  similar  manner, 


(22). 


Fig.  31. 
3 


Hence  the  square  of  a  imit  vector  is  —  1. 

The  meaning  of  the  word  "  square  "  is  more  general  than  that 
which  it  possesses  in  Algebra,  as  was  that  of  the  word  "  product" 
in  Art.  27.  The  •propriet}-  of  this  ex- 
tension of  meaning  lies  in  the  fact  that 
for  certain  special  cases,  the  processes 
above  defined  reduce  to  the  usual  alge- 
braic processes  to  which  these  terms 
were  originall}'  restricted.  The  conclu- 
j,  sion  i^  =  — 1  is  seen  to  follow  directly 
from  the  definition,  since  if  i  operates 
tAvice  in  succession  on  either  ± ,;  or  ±  fr, 
it  turns  the  vector,  in  either  case  suc- 
cessivel}'  through  two  right  angles,  so 
that  after  the  operation  it  points  in  the  opposite  direction.  A 
similar  reversal  would  have  resulted  if  the  minus  sign  had  been 
written  before  the  vector.  Thus  —  {±j)  —  '^  j.  Hence  i  x  i, 
or  P,  as  an  operator,  has  the  effect  of  the  minus  sign  in  revers- 
ing the  direction  of  a  line. 


29.  It  is  to  be  observed  that  so  long  as  the  cyclical  order  i, ./, 
Zc,  i,  J,  Zi:,  i,  ....  is  maintained,  the  product  of  any  two  of  these 
three  vectors  gives  the  third  ;  thus 


and  therefore 


as  also 


ij=k,    jk  =  i. 


ki 


(y)A'  =  Al-  =  /r=-l, 
(jJc)i=  a  =  P  =  -h 


i{jk)=  a 
ji^ci)=jj 

k{ij)=kk 


r=-l, 
A--=-l, 


GEOMETRIC    MULTIPLICATION    AND    DIVISION.  43 

hence 

;(/.-/)  =  O'A-)^ 
A-(//)  =  (7.-0J, 

which  involves  the  Associatice  law. 

We  may  therefore  omit  the  parentheses  and  write 

ijlc  =  jld  =  m=-l (23), 

or,  the  continued  product  of  three  rectangular  unit  vectors  is  the 
same  so  long  as  the  cyclical  order  is  maintained. 

But 

l-Ui)=k{-h)=^-lr=l     ....     (24), 

or,  a  change  in  the  cyclical  order  reverses  the  sign  of  the  product. 

30.  In  Equation  (24)  we  have  assumed  that 

/,(_;,)  =  _/,/,. 

That  this  is  the  case  appears  from  the  fact  that  /  operating  on 
—  /  produces  —  *,  or 

i{-J)  =  -l<: 

and  that  the  same  result  would  be  obtained  l\v  operating  with  i 
on  j,  producing  k,  and  then  reversing  k.  That  is,  to  turn  the 
negative,  or  reverse,  of  a  vector  through  a  right  angle,  is  the 
same  as  turning  the  vector  through  a  right  angle  and  then  re- 
A-ersing  it.  The  negative  sign  is,  tlierefore,  comnudative  v-ith  i, 
j.  k,  or 

i{-J)  =  -U  =  -J^ (2^)- 

31.  It  follows  directly  from  the  definition  of  multiplication. 
as  applied  to  rectangular  unit  vectors,  that  the  commutative  prop- 
erty of  algebraic  factors  does  not  hold  good.     For 

but 

Ji  =  -k. 


44 


QUATERNIONS. 


Hence,  to  change  the  order  of  the  factors  is  to  reverse  the  sign 

ofthex)roduct.     The  operator  is  always  written  first ;  and,  since 

the  order  cannot  be  changed  without    affecting    the    result,  in 

reading  such  an  expression  as  ij  —  A:,  tliis  sequence  of  the  factors 

must  be  indicated  by  saying  "• »  into  J 

^is-^^-  equals  k"  and  not  "  i  multiphed  by  J 

equals  Ic"  the  latter  not  being  true. 

Hence  also  the  conception  of  a  quo- 
tient as  a  factor  requires  a  similar  dis- 
tinction, which  in  Algebra  is  unneccs- 
c 


sarv.      In   the    latter,   from 


have,  indiftereutly,  ab  ■ 


But  from   -  =  ^ 


•hik 


=  c  and  ba  —  c. 
ij  =  h   is    true. 


//  =  7v  is  not  true.  In  expressing  therefore  the  relations  be- 
tween /,  j  and  k  by  nmltiplication  instead  of  division,  care  must 
be  taken  to  conform  to  the  definition,  the  quotient  being  used 
as  the  multiplier  or  operator  on  the  divisor.  This  non-com- 
mutative property  of  rectangular  unit  vectors,  wMch  results 
directly  from  the  primary  definition  of  the  operation  of  multipli- 
cation, will  .be  seen  hereafter  to  extend  to  vectors  in  general 
and  to  quatei-nions,  whose  multiplication  is  not  commutative 
except  in  special  cases. 

The  quotient  then  being  a  factor  which  operates  on  the  divisor 
to  produce  the  dividend,  we  have 


A-,     that  is,     ~.J 


k 


(2(5), 


the  cancelling  being  performed  l)y  an  upward  riglit-handed  stroke 
But^^-.  =  I-  is  not  true,  for  this  would  involve  ,//  =  //. 


32.    It  follows  also  that  the  directions  of  rotation  of  a  fraction 

as  -,  and  its  reciprocal  are  opposite.     Thus 
J 


k 


(27), 


GEOMETRIC    MrLTlPMCATlON    AND    DIVISIOX.  45 

and  llu'ivfore  that  the  ivciprocal  of  the  (inoticnt  i  is  —  <',  or 

\  =  -- C:^'^); 

that  is,  the  reciprocal  of  a  unit  vector  is  the  vector  reversed.     Tliis 

may  be  Avritton 

\^i-^  =  -i (21)), 

I 

the  exponent  denotuig  that,  as  a  factor  or  versor,  i  is  used  once, 
while  the  minus  sign  before  the  exponent  indicates  a  reversal  in 
the  direction  of  rotation. 

33.    If  a  be  an}-  unit  vector,  we  obtain  from  the  preceding 
Article 

..1 =.(-.) 

=  _  aa  =  -  a-  =  1 . 

But 

/ 

hence 

-a  =  ai (;50), 

a  a 

or,  a  unit  vector  and  its  reciprocal  are  commutatice  and  their 
product  2iltis  unity. 

If  a  is  not  a  unit  vector, 

a  =  T«ra, 

i=-i-=._^Ua (:n), 

a      TaUa  Ta 

the  tensor  of  the  reciprocal  of  a  vector  being  the  reciprocal  of  its 
tensor. 

It  must  be  carefully  observed  that  a  fraction,  as  4-  cannot  be 

written  inditferentlv  A'-  or  -A',  for  tliis  would  involve  ki~'^  =  i~^k, 

i        i 
which  is  not  true. 


46 


QUATEENIONS. 


B\^  definition  k  (—  i)  =  —  /,  or  ki  ^  =  k-  =  —  j  =  -.     Hence, 

i  '        i 

From  the  nieaning  attached  to  the  ordinary 

notation  of  algebra, 

(«) 


A-      ,  1         7 
-  =  k-  or  A 


A'. 
-I 


wonld   appear   to   be   correct ;  for,  cancelling,  we  have  A;  =  A' 

Whereas,  since  -  must  be  written  A-,  we  should  have 
?  i 

iki-'  [=  -  iki']  =  ki-'  i  [=  k] 
or 

;i[=  — A-]=  A;, 


whicli    is    not   true. 


Of  course  that  equation  (a)  is  .false  is 
directly  evident  from  the  fact  that 
-=  — /,  and  (a)  involves  i  (  — /)  =  (  —j)  i 

or  ij=ji.  The  above,  however,  shows 
that,  as  cancelling  must  be  performed 
b}'  an  upward  right-handed  stroke 
when  the  expression  is  in  the  form  of 
a  quotient  or  fraction,  so  when  ex- 
pressed in  the  form  of  multiplication, 
the  cancelled  factors  must  be  adjacent. 
In  such  an  expression  as 


j>   >.r  -jj 


(&) 


it  might  be  supposed  permissible  to  write  also 


(0 


since  in  either  case  the  coi-rect  result  is  obtained.  This  arises, 
however,  from  the  fact  that  both  the  fractions  in  the  first  mem- 
l)er  of  (b)  ai-e  equal  to  k.  and  therefore  may  be  permuted  so  as 


to  read  A'A- 


1.     The  process  of  (c)  is,  how- 


GEOMETRIC    MULTIPLICATIOX    AND    DIVISION.  47 

ever,  illeiiitiiiiatc.  and  tlie  result  is  correct,  not  l>ecause  the 
process  is  so,  but  bt'cause  the  factors  are  in  this  case  couiuui- 
tative. 

34.  Since  the  act  of  tension  is  indi'ijcndent  of  that  of  version, 
and  their  order  is  innnaterial, 

xi  .  >/J  =  .mj  .  //  =  yx  .  !J  =  zk       .      .      .     (32) , 

where  x  and  y  are  any  two  scahus  and  xy  =  z.  Hence  the  com- 
nuitative  i)rinciple  applies  to  tensors.  If  then  a,  /?,  y  are  in  the 
direction  of  i,  j  and  A,-  respectively,  and  a,  h^  c  are  their  tensors, 

a/3  =  TaTfS  .  ij  =  ah  .  I; 

ay  =  TaTy  .  )k  =  —  ((c  .  J,  etc.. 

or,  the  product  of  any  two  rectangular  vectors  is  the  product  of 
their  tensors  and  a  unit  vector  at  right  angles  to  their  plane. 
So  also 

g  ^  T.x  .  /  ^  Ta        i^_'_ij. 
;8~Ty3.j"T/3   *   ./"       b'' 
a  _Ta  .  i  _a  .      .  ^ 
y^TTTl-'^  c"''  ^  ^'■' 

or,  the  quotient  of  two  rectangular  vectors  is  the  quotient  of  their 
tensors  times  a  unit  vector  at  right  angles  to  their  i)lane. 

35.  If,  as  above,  a  =  ai\  then 

aa  =  ai .  ai, 

a-i=  (i-  /-, 

a'  =  -<r (.13). 

Hence,  the  square  of  any  vector  is  mimis  the  square  of  its  tensor. 
Since  Ta  =  a  is  the  ratio  of  the  lengths  of  a  and  Ua,  the  square 
of  any  vector  is  the  square  of  (he  correspondim/  line,  regarded  as 
a  length  or  distance  only,  with  its  sign  changed. 
If  ai  =  a  and  hi  —  (3, 

a/3  =  obi'-  —  —  ab. 


48 


QUATERISriONS. 


36,  That  the  multiplication  of  rectangular  vectors  is  a  dis- 
tributive operation  maj'  be  seen 
directl}'  from  Fig.  32  b^-  ob- 
serving that 


iU  +  ^)  =  ii  +  ik    (34). 


Fig.  3-2 


i  being  perpendicular  to  and  in 
front  of  the  plane  of  the  paper. 


37.    Exercises  in  the  transformations  of  i, ./,  k : 

1.  j(-i)  =  k.  2.  ./(-A-)  = 

3.   A:  (-./)=/.  4.  ^(-0  = 

5.   -A- (/)=/.  6.  {-A-)i  = 

7.  (-./)A-  =  -^  8.  (-,/)(_  A')  = 

9-  (-./)(-'•)=  10.  (-0(-./)  = 

11.  ^  =  - A.  12.  ^= 


13. 
1."). 
17. 
19. 
21. 


23.   i^ 


—  ■/ 
-A 

i 
ik 


14. 


IC.    ~  =  J. 
J 

18.^= 


20. 


22. 


24. 


1     A- 


J      « 


25.    Is  it  correct  to  write,  in  general,  the  product  of  an}-  frac- 

i-  A'     /    .     ,,      ,.         ki ., 

tions,  as  -  .  -,  m  the  lorin  —  i 

J    J  JJ 


2C).    State  whether 


k    '  i  ' 

i'rk'  =  -{ijky. 


-i^  IS  correct  or  not,  and  whv 


GEO^Nnrruic  iMiiL'rii'i.ujA  tion  and  division.        49 

38.    Ixrsuiniiig  Eciuutioii  (1"»), 

a       Ta 
n  =z  -  =  —  (cos  4>  +  « «in  4>)  '< 

tiio  quati'niioii  7  was  shown  (Art.  'io)  to  be  tlic  product  of  a 
,tcnsor  mul  a  versor.  It  ma}-  also  l)o  regarded  as  the  siiui  ol"  two 
parts,  the  first  of  Avhich  —  cos  c;^  is  a  scahir,  whose  sign  is 
that  of  tlie  cosine  of  the  angle  (c^)  between  the  A'ectors.  while  the 
second    —  sin  d)  .  e     is  a  vector  at  right  ani>les  to  their  i)lane, 

It/3      ^     J  '        ' 

whose  sign  depends  upon  the  direction  of  rotation  of  flic  Iraction 

-.     This  may  be  exijressed  synibolicallv  in  the  notation 
13  .  i  .  . 

^      (3         [3         f3 


so  that  we  have  both 
and 


q  =  TqVq 
q  =  Sq  +  yq. 


The  second  member  of  this  last  equation  is  read  "  scalar  of  q 
plus  vector  of  q."  Sry  and  Vg  being  respectivel}'  symbols  for  the 
scalar  and  vector  parts  of  the  quaternion.  As  already  explained 
in  the  case  of  the  sj-mbol  S,  V  is  a  sj'mbol  of  operation,  denoting 
the  operation  of  taking  the  vector  terms  of  the  expression  before 
which  it  is  written. 

The  quotient  of  two  vectors  is,  therefore,  the  sum  of  a  scalar 
and  a  rector. 

The  scAthir  of  the  quotient    Sg  =  —  cos  ^     is  the  ratio  of  the 

tensors  times  the  cosi)ie  of  the  contained  angle.      The  tensor  of 
the  vector  part     TWy  =  —  sin  0     is  the  ratio  nf  the  tensors  times 

the  sine  of  the  contained  angle.      The  versor  of  the  vector  part 
[UVg  =  e]  (".s  a  unit  vector  perpendicxdar  to  their  plane ^  having  a 


Of  T)^E 

UNIVERSITY 
^      Of       ^ 


50 


QUATERNIONS. 


direction  such  that  the  direction  of  rotation  of  the  divisor  is  posi- 
tive or  left-handed. 

Letting  a  and  b  be  the  tensors  of  a  and  ^8,  and  collecting  the 
preceding  expressions  for  facility  of  reference,  Ave  have 


Tq-- 


Sq 
Yq 
T\q 
VYq 
SVq 
TVq 
TVUry 


Uq'  =  cos  <^  +  e  sin  (f> 

a 


-y  s\ncf> 

0 

a    .     , 

■■  -  SUl  (f) 


■■  sin  (f> 
■  sin  <fi 


(30). 


These  expressions  require  no  further  explanation  than  that 
derived  from  a  simple  inspection  of  Equation  (15)  in  connection 
with  the  meaning  already  assigned  to  T,  U,  S  and  V  as  symbols 
of  operation. 

39.    De  Moivre's  Formula. 

The  following  considerations  will  explain  why  the  parenthesis 
(cose^  +  csinc^)  as  a  versor  turns  ft  left-handed  through  an 
angle  cf>.  They  also  contain  the  quaternion  intt'rpretation  of 
imaginar}'  quantities. 

Let  V  =  sin  ^  and  z  =  cos  4>. 

DifFerentiatins, 


dv  =  cos  <^  d(ji,     dz  =  —  sin  (f)  d(f}, 

dr  =  zdcf), 
dz  =  -  r(U. 


(a) 
(&) 


=  (.?<^.V-1,  (c) 


GEOMETRIC    MrLTll'LlCATION    AND    DIVISION.  51 

Multiplying  (a)  b^y  V— 1.  and  adding  tlie  result  to  (b), 

dz  +  (Ir  .  V^=  {-V  -{-z  V^  )  (14, 

or  

dz  +  dr  .  V^  =  ( y  V-  1  +  z)  V  -  1  dcf>, 
wlienee 

2  +  V  V  —  1 

wliic'h  may  be  written  

z  -j-  r  V— 1=  e'*^^, 
or 

eos<^  +  sin</)  .  V-l  =  e»-^^,  (^) 

whence  __^ 

cos ?i?.c/)  +  sin m^  .  V  —  1  =  e"'^'>^i.  (e) 

But  we  ha,ve  from  (d) 

(cos(^  +  sinc^  .  \f^)m=em'i^~l,  (/) 

and  therefore,  from  (o)  and  (/), 

(cos</)  +  sin</)  .  v'^)"'=cosw^  +  sin  ?«^  .  V— 1    (37). 

which  is  the  well-known  foi-mula  of  Do  Moivre. 

This  formula  may  be  made  the  basis  of  a  system  of  analyticiil 
trigonometry.  Thus,  for  example,  to  deduce  the  formidae  for 
the  sine  and  cosine  of  the  sum  of  two  angles,  we  have  from  ('0 

cos^  +  sin<^ V— 1  = ''   ^2.'' 
cos^  -f  sin  ^V^  =«""'"'• 

Multiplying  memlxn-  l)y  member, 

cos  ()>  cos  0  -+■  cos  (f>  sin  ^  .  V  —  1  +  cos  ^  sin  <^  .  V  —  1  — 

sin</,sin.9  =  .^'-'^"^^^.  (f/) 

But  from  De  Moivre's  formula 

cos  (c/>  +  ^)  +  sin  (<^  +  ^)  V^=  e^"^"^-""^-  ('0 


52 


QUATERNIONS. 


Eqiuiting  the  first  members  of  (f/)  and  (/;),  since  in  any  equa- 
tion between  real  and  imaginary-  quantities  these  are  separately 
equal  hi  the  two  members,  we  have 

cos  (9  -\-  (fi)  —  cosO  cos  (f)  —  sin  0  sin  <f). 
sin  (9  -^  cf))  =  sin  9  cos<^  +  cos^  sin  (ft. 


These  formulae,  while  they  may  be  of  course  demonstrated 
independently  of  De  Moivre's  formula,  are  here  deduced  from 
imaginary  expressions.  It  would  therefore  appear  that  these 
expressions  admit  of  a  logical  interpretation. 

If  an}-  positive  quantity'  -m  be  multiplied  by  (V— 1)'  the  re- 
sult is  — VI.  That  is,  in  accordance  with  the  geometrical  inter- 
pretation of  the  minus  sign,  we  may  regard  the  above  factor 
( V— 1)-  as  having  turned  the  linear  representative  of  vi  about 
the  origin  through  an  angle  of  180?  If,  instead  of  multiplying 
Qii  by  (V  — 1)^,  we  multiply  it  by  V  — 1,  we  may  infer  from 
analogy  that  the  line  m  has  been  turned  through  an  angle  of  90° 
about  the  origin.  If,  too,  we  ob- 
'^'  ■ '  ■  serve  that  each  of  the  four  expres- 


I' 

/ 

'il\^    \ 

X'\    -,r, 

0      m 

Uv 

\ 

-mV^l/ 

^"- 

r' 

v^. 


■')», 


v^ 


is  obtained  from  the  preceding  by 
multiplying  by  the  factor  V  —  1 ,  they 
may    be    regarded   as   denoting    in 
order  a  distance  m  on  the  co-ordi- 
nate   axes    OX,    OY,    OX,'    OY' 
(Fig.  33),  V— 1  being,  as  a  factor, 
a  versor  turning  a  line  left-handed  through  a  quadrant.     These 
expressions  therefore  locate  a  point  on  the  axes,  both  as  to  dis- 
tance and  direction  from  the  origin. 

Since  every  imaginar}-  expression  can  be  reduced  to  the  form 
±  a  ±fj  V  —  1,  we  may,  in  accordance  with  the  above  interpre- 
tation of  V— 1,  regard  such  an  expression  as  defining  the  posi- 
tion of  a  point  oict  of  the  axes.     Thus  oa  =  a  (Fig.  34)  and 


GEOMETRIC    JMULTIPLICATION    AND    DIVISION. 


53 


p'\ 


AP  =  h,  laid  off  at  a  at   ri,tilit  angles  to  oa  since  l>  is  miilliplicd 
\yy  V^  ;  so  that  in  passing  over  oa  and  Ar  in  succession  we 
reach  the  point  r.     It  is  also  evident  that  such  an  expression 
implicitly  fixes   the  position  of  i*  by 
polar  co-ordinates,  since  Va-  +  b-  =  or  '°' 

and    tanpoA=    .       In    like    manner 

(I 

_/^-j-,,V  — 1  would  locate  a  point  v', 
oa'  having  a  length  =  a,  but  laid  off 
perpendicular  to  oa,  since  V— 1  is  a 
factor,  and  a'p'=  —  I>.  As  before, 
we  have  implicitly  op' =  V(r' +  6'-  and 

tan  p  OA  = . 

b 

Furthermore,  if  w^e  op.'rate  on  the 
first  expression,  o  +  ^V  — 1,  which  fixes  the  point  p,  with 
V— 1,  we  obtain  the  second,  — /v  +  aV  — 1,  or  V— 1  as  a 
factor  turns  op  through  90°  so  as  to  make  it  coincide  with 
op!  As  an  operator,  therefore,  we  may  regard  V  — 1,  like  /, ./. 
Jc,  as  a  quadrantal  versor,  turning  a  line  through  a  quadrant 
in  a  positive  direction.  Algebraically  it  denotes  an  impossible 
operation.  (In  Algebra  quantities  are  laid  off"  on  the  same 
line  in  two  opposite  directions.  +  and  — .  It  was  because  quan- 
tities are  so  estimated  only  in  Algebra  that  vSir  W.  Hamilton 
called  it  the  Science  of  Pure  Time,  since  time  can  be  estimated 
only  into  the  future  or  the  past.)  But  it  is  unreal  or  imaginary 
only  in  an  algebraic  sense.  If  the  restrictions  imposed  by  Al- 
gebra are  removed,  by  enlarging  our  idea  of  quantity  and  at  the 
same  time  modifying  the  operations  to  which  it  is  subjected,  this 
imaginary  character  disappears.  In  a[)plying  the  old  nomen- 
clature to  these  new  modifications,  it  will  be  seen  that  the  prin- 
ciple of  permanence  is  observed,  i.e.,  the  new  meaning  of  terms 
is  an  extension  of  the  old  ;  and  when  the  new  comi)lex  quantities 
reduce  to  those  of  Algebra,  the  new  operations  become  identical 
with  the  old. 

If  now  we  operate  upon 

a  +  6  V  — 1, 


64 


QUATERNIONS. 


which,  if  we  regard  a  =  ox  (Fig.  35)  and  ftV  — 1=ap  as 
vectors,  is  equivalent  to  op.  with  tlie 
expression 

cos  </)  +  sin  (^  .  V  —  1 


Fig.  35. 


Y 


^^-^.i!. 


of  De  Moivre's  formida,  we  obtain 

a  cos  ({>  —  b  sin  (^  +  V  —  1  (a  sin  ^  + 
b  cos(^). 

Draw  OX'  so  that  X'OX=cf>;  also 
pa"  and  al  perpendicular  and  as  par- 
allel to  OX'.     Then 


■  A''  a  cos  cji  —  b  sin  <^  =  ol  —  a"l  =  oaJ' 

a  sin  ^  +  Z>  cos  (^  =  la  +  sp  =  a"p. 

Make  oa'=  oa,"  and  lay  off  a'p'=a"p  perpendicular  to  OX, 
since  it  has  V— 1  as  a  factor  ;  then 

(a  cos  4>  —  b  sin  </>)  +  V  —  1  (a  sin  ^  +  &  cos  </>)  =  oa'-|-  a'p'=  op,' 

and  f'of  —  (ji. 

But  the  formulae  for  passing  from  a  set  of  rectangular  axes 
OX,  0  Y,  to  another  rectangular  set  OX',  O  Y',  are 

X  =  a;'cos  ^  +  ?/'sin  c/j, 
?/  =  y'cos  ^  —  a;'sin  ^, 

in  which  XOX'=  <f),  .r  =  OA,  ?/  =  ap,  ;«'=  oa,"  ?/'=  pa,"  or 

OA  =  OK  +  KA, 
AP  =  NP  —  a"k, 


a"k  being  perpendicular  and  a"n  parallel  to  OX. 

Hence  the  effect  of  the  operator  has  been  to  turn  op  left- 
handed  through  an  angle  ^,  which  is  equivalent  to  turning  the 
axes  right-handed  through  the  same  angle. 


GEOMETRIC    MULTIPLICATION    AND    DIVISION.  .'>0 

+  1,-1  and  V  — 1  are  particular  cases  of  the  general  versor 

cos  (f)  +  sin  <^  .  V  —  1 , 

namel}-,  when  ^  is  0°  180°  and  !»0°  respectively,  +  1  preserv- 
ing, —  1  reversing  and  V— 1  semi-inverting  the  line  operated 
upon. 

AVe  niav  now  see  the  meaning  of  De  INIoivre's  formula 

(coS(f)  -f  sin^  .  V— 1)'"  =  eosmcf)  +  sin  mcfi  .  V  — 1. 

As  operators,  the  first  member  turns  a  line  through  an  angle  eft 
successively  m  times,  while  the  second  member  turns  it  m  times 
this  angle  once,  producing  the  same  result.  The  expressions 
cos  <^  +  sin  <^  .  V  — 1  and  cos  </>  +  sin  </> .  e  are  identical,  except 
that  in  the  latter  the  plane  of  rotation  is  not  indeterminate, 
being  perpendicular  to  c,  V  — 1  being  any  unit  vector  icith  in- 
determinate direction  in  sjxtce. 

Equation  (37)  may  be  put  under  the  form 

cos?»  {2 -It  +  c/))  +  sin//;  {-Iini  +  ^)  .  V  — l  =  [cos  (27rn  +  <^)  + 
sin(27rH+(^)  .  V^]'". 

In  the  second  member  if  <^  =  0  and  m  =  },  we  have  S/l  for  all 
integral  values  of  n,  while  the  first  member  for  n  =  0,  n  =  l, 
n=°2  becomes  1,  -i  +  ^#V^^l,  -^-^V-l,  the  three 
roots  of  unity. 

In  the  same  wa}'  for  m  =  |-, 

1, 

i  +  ^V^ 
1, 


■^I.^ 


the  six  roots  of  unity.     The  real  roots  lie  on  the  axis,  along 
which  direction  is  assumed  plus  and  minus,  while  the  imaginary 


56 


QUATERNIONS. 


roots  are  vectors  in  a  direction  not  that  of  the  axis,  and  are  the 
sum  of  two  vectors,  one  of  which  is  in  the  direction  of  the  axis 
and  the  other  perpendicular  to  it. 


40.  Let  a  and  fi  be  unit  vectors  along  oa  and  ob  (Fig.  36). 
Resolve  oa  =  a  into  the  two  vectors 
OD,  DA.     Then 


Fig.  36. 


But 


OA  =  a  =  OD  +  DA. 

OD  =  COS  ^  .  j3, 

e(sin<^  .  /5)  =  sin0  «  e^, 


e  being  a  unit  vector  perpendicular  to 
the  plane  aob,  as  in  the  figure.     Hence 


+  sine/)  .  e/ 


(a) 


Now  when  a  and  /3  are  unit  vectors,  we  have  by  definition 
-  .  /3  =  (cos  ^  +  esin  </))/3  =  a  ;  or,  comparing  with  (a) , 

(cos  </)  +  e  sin  <^)/3  =  cos  </)  .  y8  +  sin  <f)  .  c/?. 

TJie  distributive  km,  therefore,  ax)plies  to  the  multijjlication 
of  a  vector  by  the  scalar  and  vector  parts  of  a  quaterniQn;  for 
if  a  and  /?  are  not  unit  vectors,  the  tensors,  as  merely-  numerical 
factors,  can  be  introduced  without  affecting  the  versor  conclu- 
sion. Resolve  /3  into  the  vectors  oc,  cb,  cb  being  perpendicular 
to  OA.     Then 

OB  =  /3  =  oc  +  CB. 

But 

oc  =  cos  ^  .  a,       CB  =  —  e  (siu  (fi  .  a)  . 

Hence 

cos  (^  .  a  —  sin  0  .  £a  =  /?, 

or,  by  the  distributive  principle, 

(cos  (ji  —  sin  ^  .  e)  a  =  ;8. 


GEOMETRIC    MULTIPLICATION    AND    DIVISION.  o7 

Using  the  two  mcinlKTs  of  this  equation  as  multipHcrs  on  the 
corresponding  members  of  (a) 

(c0S(^  —  sin(;t  .  e)  aa  =  /3  (cos^  .  ^  +  sin  ^  .  e/3) , 

or,  since  cr  =  —  1 , 

—  cos</)  +  €sin<^  =  /?a (38). 

II"  (I  and  fi  are  not  unit  vectors, 

/3a  =  T^Ta  (  -  cos  </)  +  £  sin  <^)       .     .     .    (39). 

Operating  with  each  member  of  {(i)  on  y8, 

a/3  =  (cos  (f>  .  (^  +  sin  </)  .  €/3)/3 
=  cos(^  .  /3-+sin</)  .  eyS^ 
=  —  cos  </>  —  €  sin  (^ (40). 

or,  if  a  and  yS  are  not  unit  vectors, 

a/3=TaT^(-cos<^-esin<^)      .     .     .     (41). 

The  product  of  an}/  tico  vectors  is,  therefore,  a  cjuaternion, 
which,  as  before,  ma}'  lie  regarded  either  as  the  sum  of  a  scalar 
and  a  vector  or  the  product  of  a  tensor  and  a  versor.  In  gen- 
eral notation 

a/3  =  Sa/3  4-Va^  =  S7+Vr/    ....     (42). 

a(i  =  T(j.Vq (43). 

The  scalar  of  the  product  [Sa/3  =  —  TaT/3  cos  <;^]  is  the  product 
of  the  tensors  and  the  cosine  of  the  supplement  of  the  contained 
angle. 

The  vector  of  the  product  [Va/3  =  -  TaT/3  sin  <^  .  e]  has  for  its 
tensor  [TVa^=  TaT/8  sin  <^]  the  product  of  the  tensors  and  the 
sine  of  the  contained  angle,  and  for  a  versor  [UVa/3  =  — c]  a 
unit  vector  at  right  angles  to  their  plane  such  that  rotation  about 
it  as  an  axis  is  positive  or  left-handed. 


58  QUATERNIONS. 

Representing  the  tensors  of  a  and  /3  In'  a  and  b,  vre  have,  as 
in  Art.  38,  from  Equation  (41), 


Fig. 


Tg  =  ab 

\]q  =  —  cos  ^  —  e  sin  tp 
Hq  =  —  ab  cos  (fi 
Yq  =  —  ab  sin  cf>  .  e 

TVg  =  a6  sin  ^ 

UVg  =  -  6 

SUg  =  —  cos  (^ 

VUg  =  —  sin  <^  .  e 
TVUg  =  sin  </> 
(TV:  S)g  =  -tan(/)       ' 


K^-t)- 


41.    Resuming  tlie  expressions  for  the  products  and  quotients 
of  a  and  (3, 

y8a  =  T/5Ta(-cos(^  +  €sin<^),  (a) 

a/3  =  TaT/3(-cos<^-esin<^),  (6) 


a        Ta 


(cos  (^  —  €  sin  (/)) . 


we  observe 


i8"'f;8^^'^^*^"^'^""^^' 


(«) 


1st.  That  if  a  and  y8  be  interchanged  the  sign  of  the  vector 
part  is  changed.  It  is  equivalent  to  a  reversal  of  the  angle  ^, 
and  consequently  a  change  in  the  direction  of  rotation.     Hence 


UV/3a  =  e 


■UVa/3 

■uv? 


(45), 


Vector  multipUraUon  is  not  therefore  in  general  commutative. 
2d.    If  tlie  vectors  are  unit  vectors, 


2,      a/3: 


(46), 


GEOMETRIC    MULTIPLICATION    AND   DIVISION.  50 

the  piodiK't  being  expressed  also  by  a  quotient.  This  is  ol" 
course  iilways  possible,  as  appears  from  {a),  (6),  (c)  and  (r/), 
and  the  transibrmatiou  luay  be  effected  thus  : 


=  _^=l^(cos</,-.sin</>),         [Eq.  (31)] 
Ja  la 


Ta   ~Ta 

-  /5tt  =  T^STa  (cos  <^  -  e  sin  <^)  ; 
or 

(3a  =  Tj3Ta  {-coscf>-\-€ siu  <^)  . 

3d.  If  (/)  =  (>,  then  in  either  (a)  and  (h)  or  {<■)  and  (<)) 
the  vector  part  of  7  ))ecomes  zero,  and  the  (luatei-nion  de- 
grades to  a  scalar.     AVhen  </>  =  0  the  vectors  are  parallel,  nnd 

a       Ta        (I 
a)3  =  —  TaT/?  =  —  (/?>,    as   in   Art.    35;    also   -  =  —  = -,    as   in 
^  '^  /3      Tf3      b 

Art.  8.     If  at  the  same  time  a  and  (3  are  iniit  vectors  -  =  ^'  =  1 

/3      a 
[or  =  aa  -1  =  —  a-  =  1]  and  a;8  =  a-  =  —  1 ,  as  in  Arts.  33  and  28. 
Jf  then  q  he  any  quaternion  and  Yq  =  0,  the  vectors  of  lohich  q 
is  the  quotient  or  product  are  parallel. 

4th.  If  <^  =  *JO°then  in  either  (^0  find  (?0  or  (r)  and  {d) 
the  scalar  part  of  q  becomes  zero,  and  the  quaternion  degi-ades 
to  a  vector ;  and  either  the  product  or  (juotient  of  two  rectangu- 
lar vectors  is  therefore  a  vector  at  right  angles  to  their  plane. 
a/3  reducing  to   —  ahe  and  -  to  '-e,   as  in  Art.  31.     If  at  the 

same  time  a  and  fi  are  unit  vectors,  a/3  =  —  e  and  -  =  c.  as  in 
Art.  27.  '' 

If  then  q  he  avy  quaternion  and  Hq  =  0,  the  vectors  cf  vhirh  7 
is  the  cpiotient  or  product  are  pierpendicular  to  each  other. 

5th.  If  an  equation  involves  scalars  and  vectors,  the  vector 
terms  having  been  so  reduced  as  to  contain  no  scalar  parts,  then 
since  the  scalar  terms  are  purely  numerical  and  independent  of 
the  others,  the  sums  of  the  scalars  and  vectors  in  each  memher 
are  separately  equ<d.     Thus  if 

X  +  da  +  hf3  =  d-{-  y  +  a'a  -f  (/>'-  h")/3       ) 
then  [ .    (47), 

x  =  d  +  y     and     aa -{- h^3  ^  a'a -j- {h'- h") (3  ) 


60  QUATERNIONS. 

which  might  also  be  written  (Art.  38) 

S  (x  +  aa  +  b/3)  =S[d-t  II  +  a'a  +  (6'-  6")/3], 
y(a;  +  «a  +  6/3)  =  V[d  +  ^  +  a'a  +  {W-  6")^]. 

Gth.    ^  being  the  quotient  which  operates  on  a  to  produce  /?, 
a 

we  have  by  definition  q 

.     ^.^^/3 (48). 

7tli.  TYa/3,  or  a6  sin<^,  ?'.s  the  area  of  a  paraJMogmm  ivJiose 
sides  are  equal  in  length  to  a  and  b  and  j^arallel  to  a  and  /3. 
Sa;8,  or  —  a/jcos(^,  is  numerically  the  area  of  a  parallelogram 
whose  sides  are  a  and  6,  and  angle  ab  is  the  complement  of  <^. 

8th.  Since  the  scalar  symbol  S  indicates  the  operation  of 
taking  the  scalar  terms, 

Sa  =  0 (49), 

and,  for  a  similar  reason, 

Ya=a (50). 

Again,  since  a-  is  a  scalar, 

Y(aO=0     .......      (51), 

S(a2)  =  -a2 (52). 

V(a-)  may  be  written  Y  .  a-,  as  also  S(a-)  =  S  .  a%  but  these  forms 
must  be  distinguished  from  (Ya)^  and  (Sa)-,  which  latter  are 
also  sometimes  w^ritten  Y"a  and  S'a. 

9th.    Comparing  (a)  and  (6), 

Sa;8=S/3a (53), 

and 

Ya^  =  -Y/3a (54). 

Adding  and  subtracting  (o)  and  (b),  we  have  also 

a/3  + /3a  =  2  Sa/? (55), 

a/3-l3a  =  2Ya/3 (5G). 


(JKOMETRIC    MULTIPLICATIOX    AND    DIVISION.  61 

10th.  a/3  .  j3a  =  {Ha(3  +Va/3)  (Sa^  -Va/3)    [Eqs.  (o^)  and  (51 )] 
=  (Sa/3)2-  Sa^Va^  +  Sa/3Va/3  -  (  Va/?)". 

Hence 

a(3.  fta  =  {Ha(3y-{\a3y     ....      (57), 

or,  from  Equation  (44), 

a^  ./3a  =  (Ta/3)- (58). 

42.   Powers  of  Vectors. 

The  symbol  i™,  «(-  behig  a  positive  whole  number,  has  been 
seen  (Art.  '2H)  to  represent  a  quadi-antal  versor  used  m  times 
as  an  operator ;  the  exponent  denoting  the  number  of  times  i  is 
used  as  a  quadrantal  versor.  By  an  extension  of  this  meaning 
of  the  exponent.  /"'  would  naturally  repi-esent  a  versor  which, 

as  a  fa -tor,  produces  the  — th  part  of  a  (piadrantal  rotation. 
Thus  /'  pi-odiices  a  rotation  through  one-third,  and  i"^  through 
three-fiflhs  of  a  (juadrant,  respectively.  With  the  additional 
meaning  attached  to  the  negative  exponent  (Art.  32),  as  indi- 
cating a,  ri'versal  in  tlie  direction  of  rotation,  we  may  in  general 
define  /',  where  i  is  any  vector-unit  and  t  any  scalar  exponent, 
as  the  representative  of  a  versor  ichidi  toouhl  cause  any  right 
line  in  a  plane  perpendicular  to  i  to  revolce  in  that  plane  through 
an  angle  t  X  !»()?  the  direction  of  rotation  depending  upon  the 
sign  of  t.  Hence  every  such  power  of  a  unit  sector  is  a  versor, 
and,  conversely,  every  versor  vuiy  be  nprcsenfed  as  such  a 
power.  _^  0^ 

Since  the  angle  (c^)  of  the  versor  is  ^  X  ^,  we  have  t  =  —-< 
and  any  versor 

may  be  expressed 

and 


cos</)  +  csin  cf) 

cosc/)  + esin^  =  e' (50), 

cosc^  —  €sin</)  =  e~^      ....     (150), 


the  vector  base  being  the  unit  vector  about  which  rotation  takes 
place,  and  the  exponent  the  fractional  part  of  a  quadrant  through 
which  rotation  occurs. 


62 


QUATERXIOXS. 


The  operation  of  which  P  is  the  agent  is  one-half  that  of 
which  i  is  the  agent,  and  therefore  two  operations  with  the 
former  is  equivalent  to  one  with  the  latter ;  or,  as  in  Algebra, 

i^i^^i  =  n+^ (Gi), 

or,  employing  the  other  versor  form,  if  a,  /?,  y  are  eomplanar  unit 
A'cctors  so  that 

a  20 

-  =  cos  </)  +  6  sin  (fi=  e- , 


then  since 


we  have 


/3 

-  =  cos9  +  e  sin 


(cos</)  +  esin<^)  {cos  9  +  esin^)  =  cos  <^  cos 61  -f  €-sin<^sin^  -f 

e(sin^cos^  +  cos  <^  sin  ^) 
=  cos  ((fi +  $)-{-  esin  (cf>  +  0). 

The  second  member  is  the  U-,  its  angle  being  (</)+6),  and 
may  be  therefore  expressed  as  the  power  of  a  unit  vector,  and 

•2(0 +  fi) 

written  e     rf —  ;  this  exponent  is  the  sum  of  the  exponents  of 
the  factors,  or 

20    29  2(0  +  6) 

e-€-=e     - (62). 

This  is  evidently  an  abridged  form  of  notation  to  which  the 
algebraic  la^o  of  indices  is  applicahle. 

Since  £-=—1  and  therefore  €•*=!;  if  e'=— 1.  f  must  be  an 
odd  multiple  of  2,  and  if  €'=+1,  t  must  be  an  even  multiple 
of  2. 

In  either  case  the  coefficient  of  tt  in  rf)  =  -tt  is  a  whole  num- 

^      2 
ber,  and  cos<^±€sin<^  degrades,  as  above,  to  the  scalar  ±1, 
since  sin  ?«7r  =  0  Avhen  m  is  an  integer. 

If  e'  =  ±  €,  t  must  be  an  odd  numl)er ;    in  which  case  also 


GEOMETRIC    MULTirLICATION    AND    DIVISION.  63 

m  =  i,  :^,  4,  etc.,   cos»i7r  =  0   and  the  vcrsor  dcgrudcs  to  the 
veetor  ±e. 

If  the  veetor  is  not  a  unit  vector,  as  xi  =  p,  to  interpret  the 
exponent,  say  pi,  so  as  to  satisfy  the  Ibrnuda 

phnh  =  n Cj-^)' 


which  is  analogous  to  Equation  (Gl),  we  must  combine  with  the 
conception  of  rotation  tlu'ough  half  a  quadrant  an  act  of  tension 
represented  by  the  square  root  of  the  tensor  of  p.  Thus,  if 
X  =  16,  and  we  write 

then 

py  =  {\G'i'^)  (iG^i^)=iG/  =  p, 

or,  if  a-=  Vh, 

p'  =  V«  .  /=  =  V2  .  i\ 
p^f},}  =  ( V2  .  i')  ( V2  .  /^)  ( V2  .  P)  =  Vh  ,i  =  p. 

And,  in  general, 

p'=:(.rO'  =  .i-'./' (G4), 

or  the  tensor  of  the  power  is  the  power  of  the  tensor,  and  the 
versor  of  the  power  is  the  power  of  the  versor.     Symbolically 

T.p'  =  (T.)'    ....:.     (G5). 
IT.//  =  (U/.)' (GG). 

Any  such  power  (p') ,  as  the  representative  of  the  agent  of 
both  an  act  of  tension  and  version,  is  therefore  a  quaternion, 
whose  tensor  and  versor  can  be  assigned  by  the  above  rules,  and, 
conversely,  every  quaternion  can  be  expressed  as  the  jwiver  of  a 
vector,  which  quaternion  may  degrade  to  either  a  scalar  or  a 
vector  as  seen  in  the  preceding  versor  conclusions.  Hence  it 
follows  that  the  index-Iato  of  Ahjehra  is  ajiplicuble  to  the  jiowers 
of  a  quaternion. 


6J:  QUATERNIONS. 

43.  Relation  between  the  Vector  and  Cartesian  deter- 
mination of  a  point. 

If  i,  J,  k  are  three  unit  A'ectors  perpeiidiciilar  to  etich  other  at 
a  common  point,  then  the  vector  from  this  point  to  an}-  point  r 
may  l)e  written 

p=:xi-\-yj  +  zk (G7), 

in  which  .r,  ?/,  z  are  the  Cartesian  co-ordinates  of  p.  If  the  vec- 
tors are  not  mutuall}'  perpendicular  and  are  represented  by  a,  /3, 
y,  then 

p^Xa  +  l/13+Zy (68), 

in  which  .r,  ?/,  z  are  the  Cartesian  co-ordinates  of  p  referred  to 
the  oUHque  axes.  So  long  as  the  vectors  a,  (3,  y  are  not  com- 
planar,  p  refers  to  any  point  in  space. 

Since  any  quaternion  q  may  be  expressed  as  tlie  sum  of  a  sca- 
lar and  a  vector,  if  iv  be  anj'  scalar,  then 

q  =  w  +  Xa-{-yf3  +  Zy (69). 

As  composed  of  four  tei'ms,  we  observe  an  additional  reason 
for  calling  this  complex  expression  a  quaternion. 
Any  vector  equation 

p  =  a  =  Cla  -\-  bfS  +  Cy, 
involves  three  numerical  equations,  as 

x=a,     y  =  h^     z  —  c^ 

unless  the  vectors  are  complanar ;  in  which  case  we  ma}'  write 

y  =  na-\-  m/3, 
and 

p=:(^x  +  z)i)a  +  (//  +  Z}n)l3, 
(T=  (a  -f  en)  a  +  {b  +  cm)  13, 

which,  for  p  —  a,  involves  Init  two  equations 

X  +  zn  —  a  +  en,     y-\-  zm  —  h  +  cm. 


GEOMETRIC    MULTIPLICATION    AND    DIVISION. 


i];y 


Resuming  the  quadrinomial  Ibini  of  7,  when  the  eomponeut 
vectors  arc  at  right  ambles,  we  have 


Tq  =  xi  +  i/j+zk         J 


(70). 


Since  (TY^)-  =  —  (\q)-  =  or  +  ?/-  +  z-,  we  have 


T\q  =  yx-  -{-y^  +  z- 


V\q 


yq        xi  -\-  yj+  zk 


(71). 


i\q     V.X--  +  y-  +  2;' 

Also,  since  (Art.  41,  10th.) 

(Try)-  =  {^qY  -  {^qf  =  vf-  +  x"  +  y~  +  z\ 
Tq=  s/'i 
\]q  = 


TVLVy 


+  ar  +  y'  +  z' 

to  +  xi  +  yJ-\-zk 

Tq 

Sq 

Vw-  +  or  +  y'  +  z' 

10 

Tq 

V^(;-  +  .^•-  +  /  +  2- 

T\q 

1     x-+y--^-z' 

Tq 


(72), 


44.  The  plane  of  a  quaternion  has  been  already  defined  as  the 
plane  of  the  vectors  or  a  plane  parallel  to  them.  The  axis  of 
a  quaternion  is  the  vector  perpendicular  to  its  plane,  and  its 
angle  is  that  included  between  two  co-initial  vectors  parallel  to 
those  of  the  quaternion.  If  this  angle  is  90°  the  quaternion  is 
called  a  Right  Quaternion.  Any  two  quaternions  having  a 
common  plane,  or  parallel  planes,  are  said  to  be  Complanar. 
If  their  planes  intersect,  they  are  Diplanar.  If  the  planes  of 
several  quaternions  intersect  in.  or  are  parallel  to,  a  common 
line,  they  are  said  to  be  CoUinear.  It  follows  that  the  axes  of 
coUinear  quaternions  are  complanar,  being  peri)endicular  to  the 
common  line.     Complanar  quaternions  are  always  coUinear,  and 


66  QUATERNIONS. 

coniplanar  axes  correspond  to  eolliiiear  quaternions,  but  the  lat- 
ter may  of  course  be  diplanar. 

Let  — —  and  — —  be  any  two  quaternions.     If  coniplanar,  thev 
o'b  o"d  -^1  L  ■>        J 

may  be  made  to  have  a  common  plane  ;  and,  if  diplanar,  their 
planes  will  intersect.  In  the  former  case  let  oe  be  an}'  line  of 
their  common  plane,  or,  in  the  latter,  the  line  of  intersection 
of  their  planes.  Now,  without  changing  the  ratios  of  their  vec- 
tor lengths,  the  planes,  or  the  angles  of  the  given  quaternions, 
two  lines,  of  and  og,  may  always  be  found,  one  in  each  plane, 
or  in  their  common  plane,  such  that  with  oe  we  shall  have 


OB        OE  CD        OE 

and,  therefore,  any  two  quaternions,  considered  as  geometric 
fractions,  can  be  reduced  to  a  conniion  denominator ;  or,  in  the 
above  case 

o'a        o"c  _  OF        OG        OF  +  OG 
o'b        o"d        oe        OE  OE 

Moreover,  a  line  oh,  in  the  plane  ao'b,  may  always  be  found 
such  that 

o'a      oe 
o'b      oh' 


and  therefore 


and 


0"C       o'a  _  OG  OE         OG 

o"d     o'b      oe  oh  ~"  oh' 

o'a  ^  0"C  _  OF  ^  OG  OF       OE        OF 

o'b  *  0"d        OE  *   OE  OE       OG        OG 


45.   Reciprocal  of  a  Quaternion. 

The  reciprocal  of  a  scalar  is  another  scalar  with  the  same 
sign,  so  that,  as  in  Algebra,  if  x  be  an}-  scalar,  its  reciprocal  is 
,      1 

X 


GEOMETRIC    MULTIPLICATION    AND    DIVISIOX. 


G7 


The  reciprocal  of  a  vector  has  been  defined  (Art.  33),  so  that, 


if  a  I>e  anv  vector,  -  =  a"^ 


Ua. 


The  reciprocal  of  a  quaternion  has  also  been  dehned  (Art. 
26)  ;  thus  ^ 

r" 

being  any  quaternion, 

P      l_..-i 


is  its  reciprocal.     The  onl}-  difference  between   the   quotients 
-  and  -  (Fig.  37)  is  that,  as  opera- 

ji  a  Fig.  37. 

tors,  one  causes  (i  to  coincide  with  a, 
while  the  other  causes  a  to  coincide 
with  y8.  A  quaternion  and  its  recipro- 
cal have,  therefore,  a  common  plane 
and  equal  angles  as  to  magnitude, 
but  opposite  in  direction ;  that  is,  ^- 
their  axes  are  opposite.     Or 


Since 


/.(I     and     axis  -  = 


axis  q. 


,:l  =  ^^:l  =  ^J  =  f^,    and    ^.^  =  ^=i 


the  product  of  tico  reciprocal  quaternions  is  equal  to  positive 
unity,  and  each  is  equal  to  the  quotient  of  unity  by  the  other; 
we  have,  therefore,  as  in  Algebra,  -q=l  and  7=  ,  and  no 
new  symbol  is  necessary  for  the  reciprocal.  -  is,  however, 
sometimes  written  Rg,  R  being  a  general  symbol  of  operation, 
namely,  that  of  taking  the  reciprocal.  It  follows  from  the  above 
that 

„1        1 


IVy 


(73), 


68 


QUATERNIONS. 


or,  the  tensors  of  reciprocal  quaternions  are  reciprocals  of  each 
other  ;  while  the  versors  differ  only  in  the  reversal  of  the  angle. 
If  then 


Ta 


q  =  -  =  —  (cos  ch  +  e  sin  (h) 


we  shall  have 


^  =  ^(co.s<^-csinc/,)    I 
a        Ta  ) 


•      {'^)' 


46.   Conjugate  of  a  Quaternion. 

If  y3'  (Fig-  37)  be  taken  coniijlanar  with  /?  and  a,  and  making 
with  a  the  same  angle  that  /3  does, 
T,3'  being  also  equal  to  T/?,  then,  if 
is  called  the  conjugate  of 


Fig.  37. 


■:\  =  ^h 


f^' 


cj,  and  is  written  Kq.     The  symbol  K 
indicates  the  operation  of  taking  the 
conjugate.    A  quaternion  and  its  con- 
^^'il(^  jugate    have,    therefore,    a    common 

plane  and  tensor,  as  also,  in  the  ordi- 
nary- sense,  equal  angles  ;  but  their  axes  are  opposite  ;  or 


Z  Kg  =  Z  (/ 
TKg  =  Tg  = 


and 
If  then 

we  shall  have 


T. 


axis  Kg  =  —  axis  g  =  axis 


9  =  ^  =  ,j7^  (cos  <^  +  esin<^) 


K</ =  ;^  ( cos </>-£ sine/)) 


(75). 


(70), 


or,  the  tensors  of  conjugate  qnaternions  are  equaU  and  the  versors 
differ  only  in  the  reversed  of  the  angle. 

Regarding  a  scalar  and  a  vector  as  the  limits  of  a  quaternion 


GEOMETRIC    MULTIPLICATION    AND    DIVISION.  G9 

(Art.  41,  3d  and  4tli),  we  see  from  Equation  (7G)  that  the  con- 
jugate of  a  scalar  is  the  scalar  itself,  and  that 

Ka  =  — a  =  — TuUa (77), 

or.  the  conjiKjdtc  of  a  vector  is  the  vector  reversed.     In  general 
notation  mo  may  write 

whence  it  follows  from  the  above  that 


Kq  =  Sq-\q  ) 

or  (Art.  43)  '■r      ....     (78), 

Kq  =  ic  —  xi  —  yj—  zk  ) 


that  is,  the  scalar  of  the  covjucjcde  of  a  quaternion  is  the  scalar 
of  the  quaternio7i,  and  the  vector  of  tlie  conjugate  of  a  quaternion 
is  the  vector  of  the  quaternion  reversed  ;  a  result  which  may  be 
expressed  symbolically 

^K^  =  ^^/      1 (79). 

These  are  Equations  (53)  and  (o4). 

If  we  add  and  subtract  the  two  conjugate  quaternions 

q  =  87  +  V7.     K7  =  S7  —  V7, 

we  have 

7  +  K7  =  2S7| (go)_ 

7  -  K7  =  2  V7  \  ^     ' 

The  sum  of  tivo  conjugate  quaternions  is,  therefore,  cdways  a 
scalar,  positive  or  negative  as  the  Zryis  acute  or  obtuse.  If 
Z  7  =  ^,  tliis  sum  is  evidently  zero. 

Since,  if  7  is  a  scalar,  K7  =  7,  then,  conversel}",  if'Kq  =  q.  7 
is  a  scalar. 


'0 


QUATEENIONS. 


47.    Opposite  Quaternions. 

If,   for  ^,  we  write   ^   (Fig.  37),  the  latter  is  called  the 
Opposite  of  q,  and  is  evidently-  —  q,  for 


g  _^  0  -  g  ^  0_a 


0-q 


As  appears  from  the  figure,  opposite  quaternions  have  a  com- 
mon plane  and  tensor,  supplementary  angles  and  opposite  axes  ; 
or 

T  (  —  g)  =  Tr/,  Z—q  =  7r—Zq     and  axis  (  —q)  =  —  axis  q. 


Since 


—  g  1   <^  _  g  —  g  __  0  ^ 


the  sum  ofhvo  opposite  quaternions  is  zero,  or 


Fig.  37. 


Q  +  (-q)  =  o. 


Also,  since 


/3      /3       /3 
^1 


1, 


or,  tlxeir  quotient  is  ner/ative  unit}/. 


If  then 
we  shall  ha's^e 


(81). 


If  ^  ^  =  ^5  Kq  =  —  q;    and,  conversely,  if  Kq  =  —  q,  q  is  a 
vector. 


GEOMETRIC    MULTIPLICATION    AND    DIVISION. 


48.  Sinco  Vq  is  independent  of  the  vector  lengths,  and  only 
dependent  upon  relative  direction,  versors  are  equal  whose  axes 
and  angles  are  the  same.     Hence 

1 


But  (Art.  24) 


U    1 


UKf/=U: 


V^  =  ^=l:^ 


and.  Equation  (82). 


.-.  II 


UKry 


Vq 


V^ 


(82). 


(83), 


Again,  since  the  conjugate  of  a  versor  is  the  same  as  the  re- 
ciprocal of  that  versor,  we  have,  from  Equations  (82)  and  (83), 


UKg  =  KVq 


(84). 


Fig. 


49.   Representation  of  Versors  by  spherical  arcs. 

If  a,  fS,  y,  are  co-initial  unit  vectors,  their  extremities  will 

all  lie  on  the  surface  of  a  unit  sphere  (Fig.  38).  ^  being  any 
quaternion,  U  .3  turns  ;8  from  the  position 
OB  to  OA,  and  this  versor  may  be  repre- 
sented by  the  arc  ii.\  joining  the  vector 
extremities  ;  for  this  arc  determines  the 
plane  of  the  versor  as  also  the  magnitude 
and  direction  of  its  angle,  the  direction 
of  rotation  being  indicated  by  the  order 
of  the  letters  as  in  the  case  of  vectors. 
This  representation  of  versors  by  vector 
arcs  is  of  importance  in  the  theorems  re- 
lating to  the  multiplication   and  division  of  (inaternions,   and 

may  be  made  upon  a  xinlt  sphere  ;  for,  if  a,  /3,  y, are  not  unit 

vectors,  the  quaternions  will  differ  from  the  versors  by  a  nu- 
merical factor  only,  the  introduction  of  which  cannot  affect  the 


72 


QUATERNIONS. 


versor  conclusions.     Disregarding,  then,  the  tensors,  since  ver- 
sors  are  equal  whose  planes  are  parallel  and  angles  equal  (in- 
cluding   direction),    equal    arcs    on   the 
Fig.  38.  same  great  circle  and  estimated  in  the 

c  same  direction  represent  equal  versors, 

for  any  arc  ma}'  be  slid  OA-er  the  great 
circle  on  which  it  lies  without  change  of 
length  or  reversal  of  direction.     On  this 
plan  b'a  =  AB  will  represent  the  recipro- 
cal or  conjugate  of  ba,  and  a  quadrantal 
versor  would  have  for  its  representative 
BC,  an  arc  of  90?     Also,  the  versors  of 
all  complanar  quaternions  will  be  rei)re- 
seuted  by  arcs  of  the  same  great  circle,  while  arcs  of  different 
great  circles  will  represent  the  versors  of  diplanar  quaternions, 
which  are  always  unequal. 

If  M,  N  and  p  are  the  vertices  of  a  gpherical  triangle,  the  vector 
arcs  MN,  NP  and  pji  will  represent  versors,  and  it  will  be  seen 
that  by  taking  the  geometric  sum  of  two  of  these  arcs  in  a  cer- 
tain order,  the  remaining  arc  will  represent  the  versor  of  their 
product ;  so  that  if  (/'  be  represented  by  pji  andg  by  np,  q'q  may 
be  constructed  by  a  process  of  spherical  addition  represented  by 
PM  +  NP  =  NM,  NM  representing  the  versor  q'q  ;  but  that  because 
q'q  and  qq'  are  not  generally  equal,  this  process  of  splierical  ad- 
dition, as  representing  versor  multiplication,  is  not  commutative 
as  was  that  of  vector  addition,  pm  +  np  and  np  +  pm  representing 
diplanar  versors. 


50.   Addition  and  Subtraction  of  Quaternions. 

Since  a  quaternion  is  the  sum  of  a  scalar  and  a  vector,  in 
finding  the  sum  or  difference  of  several  quaternions  the  sum  or 
difference  of  tlieir  scalar  and  vector  parts  ma}'  be  taken  sepa- 
rately. The  former  will  be  a  scalar  and  the  latter  a  vector; 
consequentl}',  the  sum  or  difference  of  several  quaternions  is  a 
quaternion. 


GEOMETRIC    MULTIPLICATION    AND    DIVISION.  73 

1.  r>()tli  the  assoc'iiitive  and  coniinutativc  priiuiitlcs  Itfiiitj; 
applicable  to  the  suiniiiatiou  of  scalars,  as  also  to  that  of  vectors 
(Arts.  4,  5),  they  also  hokl  good  for  the  addition  and  subtrac- 
tion of  quaternions  ;  or 

V  +  '•='•  +  'y         ) 

and  [     .      .     .     .     (So). 

<I  +  {r+s)  =  {q  +  r)  +  s) 
If  then 

r  =  Sr  +Yr 

.s  =  7+r  + =  S.S+V6-; 

in  Avhich 

S.S  =  S  (7  +  /•  + )  =  Hq  +  Sr  + , 

\s  =  y{q  +  r  + )=\q  +  Yr  + , 


and,  in  general, 


SSry  =  2S7l 
TS7  -  SVfy  i 


(8G), 


or,  in  quaternion  addition  and  subtraction,  S  aurf  Y  are  distribu- 
ticc  symbols. 

2.  If  q  +  r  -\-ji  + =  s,  then,  Equation  (78), 

Kry+  Kr  +  Ky<  + =  87+  Sr  +  Sj)  + -\q-\r-\2^  - 

=  S.s-V.s  =  lvs. 

•••   2K7  =  KS7 (87), 

K,  like  S  and  V,  being  a  distributive  sj'uibol. 

3.  Again,  since  the  conjugate  of  a  scalar  is  the  scalar  itself, 

KSf/  =  Hq. 
But  S7  =  SK7.     Hence 

KS7  =  87  =  8X7 (88). 

Also,  since  the  conjugate  of  a  vector  is  the  vector  reversed, 

KV7  =  -V7. 


74 


QUATERNIONS. 


But  —  V^  =  YKq.     Hence 

KYq  =  -Yq=\Kq (89); 

hence  K  is  commutative  ivith  S  u)td  V. 

4.  Since  an}'  two  quaternions  ma}'  be 
reduced  to  a  common  denominator  (Art. 
44),  so  that 

a      y  _  g'+y' 
/3^8~     8'    ' 
and  since 

Ta'+Ty'>T(a'+y') 

unless  a'  =  .Ty'  anda;>0,  it  follows  that 

Tq-^Tq'>T{q  +  q') 

unless  q=:xq'  and  a;>0.  Hence,  in  general,  T2g  is  not  equal 
to  STg.  Moreover,  since  TJSq'  is  a  function  of  the  tensors  under 
the  2  sign,  while  ^Vq  is  independent  of  the  tensors,  U2g  is  not 
equal  to  2Ug.  This  also  appears  from  the  representation  of  ver- 
sors  b}' spherical  arcs  (Fig.  38).  Hence,  in  the  addition  and 
subtraction  of  quaternions,  T  and  V  are  not,  in  general,  dis- 
tributive symbols.  ■ 


51.   Multiplication  of  Quaternions. 

1.    Let 

g  =  Sfy+Vg,     ?-=S)-+yr 

be  any  two  quaternions.     Then 

p  =  qr  =  SrySr  +  SgVr  +  S>-Vg  -j-YqYr. 

The  last  member,  being  the  sum  of  a  scalar  and  a  vector,  is  a 
quaternion.  Hence,  the  product  of  two  quaternions  is  a  quater- 
nion, and 

p  =  ^p  -\.Yp  =  ^qr  -{-Yqr, 
in  which 

Sgr  =  SgSr  +  S  .  Vry Vr       ....     (90), 
and 

Yqr^^qYr  +  ^rYq+Y  .YqYr      .     .     .     (91). 


GEOMETRIC   MULTIPLICATION   AND   DIVISION.  ( T) 

If  wo  multipl}-  q  by  r,  we  obtain 

yrq  =  SrYq  +  S^Vy  +  V  .  \r\q. 
But,  Equation  ("^3), 

S  .  YrYq  =  S  .  Yq\r. 
.-.   Srq  =  iiqr (92). 

But,  Equation  (54), 

Y  .YqYr^-Y  .YrYq, 

and  therefore  the  products  qr  and  rq  are  not  equal.  Hence, 
quaternion  multi'plication  is  not  in  general  commutative.  If, 
however,  q  and  r  are  complanar,  Yq  and  Yr  are  parallel,  and 
V  .  YqYr  =  0  ;  in  which  case  qr  =  rq.  Conversely,  if  qr  =  rq,  q 
and  r  are  complanar. 

Since  Reciprocal,   Conjugate    and  Opposite  quaternions   are 
complanar,  they  are  commutative,  or 

qKq  =  Kq  .  q  -. 

(j~  =  -q  =  qq-'  =  q-'q  'r     .    .    .    (93) . 
9(-9)  =  -w  -■ 

2.    It  has  been  shown  (Art.  44)   that  any  two  quaternions 

q,  q',  can  be  reduced  to  the  forms  L  and  Z  having  a  common 

a  a 

denominator,  or  to  the  forms  -  and  Z.     Hence 
S  a 


We  have  then 


,  y    p     7     «      y 

q  :  q  =  -  :  -  =  ~  '  -  =  -  • 
^  ^      u  •  a      a    (3     (3 


(/  13       TI3       Ta      T/3       Ta      Ta       *^   •   *^  ^ 


76  QUATERNIONS. 

In  a  similar  nianuer 

Hence  the  tensor  of  the  product  (or  quotient)  of  any  two  qua- 
ternions is  the  product  (or  qtiotient)  of  their  tensors,  and  the  ver- 
sor  of  the  product  (or  quotient)  is  the  product  (or  quotient)  of 
their  versors. 

In  fact,  tensors  being  eommntative,  we  have,  in  general, 

TUq  =  -n.Tq (96), 

Uq  =  TRq  .  VUq  =  UTq  .  nU(/, 

.-.   VUq  =  IlVq (97). 

3.  The  multiplication  and  division  of  tensors  being  purety 
arithmetical  operations,  we  proceed  to  the  corresponding  opera- 
tions on  the  versors.  It  has  been  shown  (Art.  44)  that  any 
two  versors  q,  q',  may  be  reduced  to  the  forms 

,  =  ^  =  55,     c/=^=^:  (Fig.  3D), 

a        OA  ^        on 

A,  n,  c^  being  the  vertices  of  a  spherical  triangle  on  a  unit 

sphere.     Then 

q'q  = 

p        a         a        UA 

If  we  represent  the  versors  q'  and  q  hy  the  vector  arcs  nc' 
and  AB,  then  the  versor  1,  the  product  of  q'q,  will  l)e  repre- 

"  y' 

sented  by  the  arc  ac' ;  moreover  if  q"  =  -    represent  anv  divi- 

iS  .  "^ 

dend  and  q=  -  an}'  divisor,  then 

q"  _  y'     a  _  y'  _  oc' , 
q  ~  a  *  jy  ~  /?  "~~  OB  ' 

the  versor  of  the  product  q'q  being 

BC'  +  AB  =  AC', 


GEOMETRIC    MrLTIPLICATIOX    AND    DIVISION.  77 

and  the  vorsor  of  the  quotient  'L- 
<I 

AV'  —  AH  =  lu'; 

and,  as  in  the  addition  and  subtraction  of  quaternions,  the  pro- 
cess consisted  in  an  algebraic  addition  and  subtraction  of  scalars 
but  a  geometric  addition  and  subtrac- 
tion of  vectors,  so  the  multiplication 
and  division  of  (juaternions  is  reduced 
to  the  corresponding  aritlimetical  ope- 
rations on  the  tensors  and  the  geome- 
trical multiplication  and  division  of 
the  versors,  the  latter  being  con- 
structed bv  means  of  representative 
arcs  and  the  rules  of  s[)lierical  addition  and  subtraction. 


Fis.  39. 


4.  The  representation  of  a  versor  by  the  arc  of  a  great  circle 
on  a  unit  sphere  illustrates  the  non-commutative  character  of 
quaternion  nuiltiplication.  For,  ab  and  ba'  (Fig.  39)  being  equal 
ai'cs  on  the  same  great  circle,  as  versors 


ab  =  I5A  , 

CB  =  BC'. 

= 

a 

a' 

and     /•  = 

7 

y' 

a 

a' 

and     r(j 

1!^ 

a 

and  similarl}- 
Now  if 

then 


the  versors  qr  and  rq  being  represented  by  the  arcs  ca'  and  ao' 
respectively.  These  arcs,  though  equal  in  length,  are  not  in  the 
same  plane,  and  therefore  the  Acrsors  rq  and  qr  are  not  ecpial. 
Constructing  these  versors,  by  spherical  adilition  we  should  have 

Bc'  -f  AB  =  Ac',' 

AB  +  Bc'  =  ba'  -f  en  =  ca', 

a  change  in  the  order  giving  uneipial  results. 


78  QUATERNIONS. 

Hence,  unless  ac'  and  ex'  lie  on  the  same  great  circle,  in 
which  case  q  and  r  are  complanar,  quaternion  multiplication  is 
not  commutative. 

5.  Other  results,  hereafter  to  be  obtained  symbolicalh',  ma}' 
be  readily  proved  b}'  means  of  spherical  arcs,  as  follows : 

If  AB  (Fig.  30)  represents  the  versor  of  q  =  -,  a'b  =  ba  repre- 

1  " 

sents  the  versor  of  Hq  or  —     Tlie   spherical  sum  of  ab  -|-  ba 

'^  1 

being  zero,  the  effect  of  the  versors  in  the  products  qKq  and  q- 

_</ 
is  to   annul  each  other.     Hence,   if  the  vectors  are    not  unit 


vectors, 

Again,  from 
we  have 


qlLq  =  Kq  .q  =  {Tqy (98). 

q       q 


AB  +  BC    =CA  , 


7 

and  the  versor  of  K  (qr)  will  therefore  be  represented  by  a'c. 
But 

a'c  =  EC  +  a'b, 

whence 

JL{qr)  =  -KrKq (99), 

or,  the  conjugate  of  the  product  of  tivo  quaternions  is  the  2^^'oduct 
of  their  conjugates  in  inverted  order. 

G.  The  product  or  quotient  of  complanar  quaternions  is  readily- 
derived  from  the  foregoing  explanation  of  versor  products  and 
quotients  as  dependent  upon  a  geometric  composition  of  rota- 
tions. For,  disregarding  the  tensors,  the  vector  ares  which 
represent  the  versors,  since  the  latter  are  complanar,  will  lie  on 
the  same  great  circle,  and  the  processes  which  for  diplanar  ver- 

a 

sors  were  geometric  now  become  algebraic.     Thus  for  q  —  -7. 

qq'=q<q=       ,'^  =  -, 
p       a        a 


GEOMETIMC    MULTl PLICATION   AND   DIVISION.  79 

iuui.  Fig.  r,;), 


also  for  q"=  '-  ixiid  q 


MX    +  AB  =  AU  +  BA    =  AA 

ft 


and 


q"  _  «/  .  ^  _  f^      «  _  «' 
(/'         a   *   u        a      fS       13 

liA  +  Aa'=  Ua'. 


The  product  or  quotient  of  any  two  complanar  quaternions  is 
therefore  obtained  by  multipbji'ug  or 
dividing  their  tensors  and  adding  or  Fig.  39. 

subtracting  their  angles.     Thus 


2Xj  =  Tp  .  T7[cos(c^  +  ^)  + 
£sin(c/,  +  ^)]. 
Up  =  q, 


q-  =  (Tq)-  (cos  2  </>  +  €  sin  2  (^)  , 

or,  generall}', 

q"=(Tq)"(co>incf)  +  e  sin  n<^) 

whence  result  the  following  general  formulae, 


v{r)={vqr    [ 

SU(5")  =  cos?iZ(/  j 
TVU(r/')  =  sin?iZ(/ J 


(100). 


(101), 


which  are  all  involved  in  Art.  42. 


52.     1.    Distributive    and   Associative  Laws    in   Vector 
and  Quaternion  Midtij>U<-ati(»i. 
Having  assumed  (Art.  24) 

a       a  a 


80  QUATERNIONS, 

whence 

since  a  is  an}'  vector,  we  have 

/?a  +  ya  =  (/?  +  y)  a.  (a) 

Taking  the  conjugate  of  {[3  +  y)a, 

K[(/3  +  7)'0  =  KaK(/3  +  y)  [Eq.  99] 

=  Ka(K/3  +  Ky).  [Eq.  87] 

Taking  the  conjugate  of  (/3a  +  ya) , 

K(^a  +  ya)  =  K/?a  +  Kya  =  KaK/3  +  KaKy. 

Hence 

Ka(K/3+  Ky)  =  KaK/3  +  KaKy, 

or 

a'(/3'+y')  =  a'/3'+aV.  (5) 

Hence,  from  (a)  and  (b),  the  mitUij^Ucation  of  vectors  is  a 
doubly  distributive  operation,  and 

(^+y)(a  +  S)  =  /?a  +  ya  +  ^S+yS      .       .      (102). 

2.  Let  7  =  y,  1>e  any  quaternion  and  a  any  vector  ;  also  /3  a 
vector  along  the  line  of  intersection  of  a  plane  j^erpendicular  to 
a  with  the  plane  of  q.  Then  another  vector,  S,  ma}'  he  found  in 
the  latter  plane,  such  that  q  —  ^i  ^  having  the  same  angle,  plane 
and  axis  as  '^.  Also  let  y  be  a  vector  in  the  intersecting  plane, 
such  that  -^  =  a.     If  now  a  he  anj'  scalar, 

^0/3 -hy     (3  ^  a  13  +  y 
=  ag  +  aq. 


GEOMETRIC    MULTIPLICATION    AND    DIVISION.  81 

Taking  the  conjugates  as  above, 

q\a'+a')  =  (ia'+<l'a'. 
Hence,  in  general, 

(a  +  a)  {il'+  a')  =  (((('+  aa'+a'a  +  aa';  {c) 

or  regarding  a,  a',  and  a,  a  each  as  the  sum  of  two  scalars  and 
two  vectors  respectively, 

(ai+  a.,  +  ai  +  oo)  (a'l  +  a'.  +  a'l  +  a',)  = 

(o,  +  a.,)  (a'l  +  "'2)  +  («i  +  ('2)  («'i  +  a'2)  +  («'i  +  "'2) 
(«i+  "2)  +  («i  +  "2)  ('^'1  +  "'2)  = 

('h  +  ai)  («'i  +  «'i)  +  (f'l  +  «i)  («'2  +  "'2)  +  («2  +  «2) 
(a'l+tt'i)  +  («2+a2)  (a'a+a'a), 

since,  from  {<-),  the  factors  in  the  expression  preceding  the  last 
are  distributive.  Putting  for  the  parentheses,  which  are  sums 
of  a  scalar  and  a  vector,  the  quaternion  symbols  2h  <h  >'  i^'^'l  ^'f' 

^'^^'""^'^  {p  +  q){r  +  H)  =  pr+ps  +  qr  +  qs    .      .     (103), 

or,  the  mnltipUmtion  of  quaternions  is  a  doubly  distributive 
operation. 

3.  Assuming  any  three  quaternions  under  the  quadrinomial 
form.  Article  43,  «,  /t,  J  being  unit  vectors  along  three  mutually 
rectangular  axes,  we  have 

q  —  iv  -\-xi  +ijj  +^A*,  (o) 

r=w'  +x'i  +i/'J  -{-z'k,  (b) 

s=iv"+x"i+u"J+z"k.  (c) 

Multiplying  first  (c)  by  (b)  and  the  result  by  (a),  and  then 
(6)  by  (a)  and  (c)  by  this  result,  observing  the  order  of  the  fac- 
toids, it  will  be  found  that  the  scalar  and  vector  parts  of  these 
two  products  are  respectively  equal,  and  therefore 

q(rs)  =  {qr)s (104), 

or,  the  associative  law  is  true  in  the  multipiication  of  quaternions. 


82  QUATERNIONS. 

53.     1.    If  a  and  /3  be  any  two  vectors,  then 

(a  +  (3){a  +  /3)  =  {a  +  /3y  =  a'  +{a/3  +  /3a)  +  ft\ 

•whence,  Equation   (^5),  or,   comparing  Equations   (39),    (41) 
and  (80), 

(a+^)-  =  a-  +  2Sa^  +  ^2        _       _       _      ^^q^^^ 

2.    Similarlv 


ia-(3){a-(S)  =  (a-/3r-: 


(a/3+^a)+yS^ 


(a-^)2  =  a--2SayS  + 


(ion), 


3.    From  Equation  (57),   or  b}'  multiplying  q  —  iiq  -{-Yq   by 

Kfy  =  Sr/  —  Jq, 

a/3./3a  =  (Sqy-(\qy-; 

hence,  from  Equation  (98),  the  equalities 

afS  .  fta  =  qKq  =  {§a/3y--{\a(3y  =  {Tqy     .      (107). 

54.   Applications. 

1.    In  any  right-angled  triangle,  the  square  on  the  hypothenuse 
is  equal  to  the  sum  of  the  squares  on  the  sides. 

Let  the  sides,  as  vectors,  be  repre- 
sented b}'  a  and  /3  (Fig.  40) ,  and  the 
hypothenuse  by  y.     Then 

7^a  +  /3. 

Squaring,  Equation  (105), 

/=a=^  +  2Sa/3+y8^ 
/  =  a2  +  /3^ 


Fig.  40. 


or.  Art.  41,  4, 


or,  as  lengths  simply,  changing  signs  [Equation  (33)], 

BA-  =  BC^  +  CA-. 

2.    In  any  right-angled  triangle,  the  medial  to  the  hypothenuse 
is  one-half  the  hypothenuse. 


GKOMETIUO    jMULTIPLICATION    AND    DIVISION.  83 

111  Fig.  40,  for  the  medial  vector  CD  =  8,  we  have  (Art.  1 ."») 

or 

2S  =  /?-a. 

8(jiiiu-iiig.  luul  since  S/3a  =  0, 

4  S-'  =  fr  +  a-, 
or 

,      CA-  +  cir      Air 

CD-  = ■ =  —  ; 

4  4 

.-.    CD  =  ^. 
2 

3.  If  the  (liagonaJs  of  a  parallelogram  are  at  ri\/ht  angles  to 
each  other,  it  is  a  rhombus. 

Let  the  vector  sides  be  represented  by  a  anil  /3.     Tlien  a  +  yg 
and  a  —  f3  are  the  vector  diagonals. 
By  condition 

S(a+^)(a-/3)=0.  [Art.  41,4] 

But,  Equation  (53), 

S(a4-/3)(a-/3)  =  a--^-"=0. 

which  is  true  only  when  Ta  =  TfS,  that  is  when  the   sides  arc 
equal. 

4.  The  Jlgure  formed  by  joining  the  middle  points  of  the  sides 
of  a  square  is  itself  a  square. 

Let  BC  and  ca  (Fig.  40)  be  the  sides  of  a  square,  p  and  q 
their  middle  points,  and  o  the  middle  point  of  the  side  opposite 
BC.     Then,  with  the  same  notation, 

PQ  =  ^(a+^),  Qo  =  ^(/3-a); 

.-.    S(PQ  .  QO)=0, 

or  PQ  and  go  are  at  right  angles. 

5.  In  any  triangle,  the  square  of  a  side  opposite  an  acute 
angle  is  equal  to  the  sum  of  the  sqicares  of  the  other  sides,  less 


84 


QUATERNIONS. 


tioice  the  product  of  the  base  and  the  line  betiveen  the  acute  angle 
and  the  foot  of  a  perpendicular  from  the  angle  opposite  the  base. 


Fig.  41. 


CD  =  a,  BA  =  y  (Fig.  41). 


-h-. 


Let  CA 
Then 

^-  =  a-  +  2Say  +  y-. 

Now 

2Say  =  -2TaTyCOs(180-B)  = 

2  ac  COS  B. 
Hence 

+  2  ad. 


r  —  c-  +  2  ac  cos  b  =  —  cr  — 
or 

b-  =  a-  +  c-  —  2  ad. 

If  B  is  a  right  angle,  Say  =  0,  and,  as  in  Example  1, 

b-  =  a-  +  c". 

"What  does  this  theorem  become  for  a  side  opposite  an  obtuse 
angle  ? 

G.    In  any  j)Jane  triangle,  to  find  a  side  in  terms  of  the  other 
tico  sides  and  their  opposite  angles. 
In  Fig.  41, 

/?=a4-y. 
Multiplying  into  a 

(3a  =  a"  4-  yo- 

Taking  the  scalars  (Art.  41,  5), 

S,ea  =  -  a-  +  Sya, 
or 

—  ba  cose  =  —  a-  —  ca  cos  (180°—  b)  ; 
.  • .  a  —  b  cos  c  +  c  cos  b  . 

The  al)ove  operation  Avith  a  is  indicated  by  saying  simpl}-, 
"  operating  with  x  S  .  a,"  meaning  that  a  is  first  introduced  and 
then  the  scalars  taken.     The  position  of  the  sign  x  will  indicate 


GEOMETRIC    MULTirLlCATION    AND    DIVISION.  85 

hoir  a  is  iisi'd.      If  used  as  jv  nuiltiplier,  we  slunild  write,  ''  ojxt- 
utiug  with  S  .  a  X  ." 

7.  The  siiioi^  of  the  avf/Jes,  in  any  plane  triangle^  are  propor- 
tional to  the  opposite  sides. 

In  Fig.  41 

Operating  with  x  V  •  a,  that  is,  as  explained  in  the  preceding 
example,  nuiltii)lying  into  a  and  taking  the  vectors  (Art.  41,  5), 

V/3a  =y(a  +  y)  a  =  V  .  a"  +Vya. 

But  V  .  a-  =  0  ;  hence 

V^a  =  Yya, 
ha  sine  =  ca  sinB, 
or 

sine  :  sin  v,::c:h. 

Notice  that  V/3a  and  Vya  involve  a  unit  vector  at  right  angles 
to  their  plane,  and  that,  owing  to  the  order  of  the  vector  factors, 
£  has  the  same  sign  in  both  members  of  the  equality,  and  may 
therefore  be  cancelled.  The  period  in  V .  a^  ma}-  evidently  be 
omitted,  as  in  V/3a ;  it  will  be  used  hereafter  only  to  avoid  am- 
biguity. Thus  Kr/r  means  the  conjugate  of  qr ;  but  Kg  .  r  is  r 
nuiltii)lied  by  the  conjugate  of  (/. 

8.  In  a  right-angled  triangle,  to  find  the  sine  and  cosine  of  the 
acute  angles. 

Let  Aii  =  y,  Ae  =  /?,  ne  =  a  (Fig.  42) . 
Then 

IS  =  y  +  a, 
whence 

Taking  the  scalars,  since  S-=  0, 
P 

c  h 

l=v-cosA,     or     eosA=-' 

b  c 


Fig.  43. 


UAT 

ERNIONS. 

Tak 

iiig  the  vectors 

1 

+^r 

=  0, 

c    . 
-  sin  A 
b 

a    . 

siu  c  = 

b 

.'. 

sin  A  = 

a 
c 

In  this  example  UV^  =  —  UV  ^• 

'J.     To  find  the  sine  and  cosine  of  the  sum  oftico  angles. 
Let  a,  y8,  y  be  complanur  unit  vectors  (Fig.  43),  and  e  a  unit 
vector  perpendicular  to  their  plane.     We  have 


Fig.  -13, 


cos(4>  +  0)  +  es'm{<{>-\-0), 


cosO  +  esin^. 


Hence 


cos ((/)  +  e)-he  sin  ((^  +  6)  =  (cos  </>  +  e  sin  </>)  (cos  ^  +  e  sin^) 
=  cos<^  cos^  4-  e{fiin<fi  cos 6  +  cos<^  sin^)  +  e'sin^  sinc^. 

Equating  the  scalar  and  vector  parts  in  succession,  there  re- 
sults, since  £- =  —  1, 

cos  (cji  -\-  0)=  cos  cf)  cos  0  —  sin  <^  sin  ^, 
sin  (</)  +  ^)  =  sin  <^  cos^  +  cos^  sin^. 

10.    To  find,  the  .s«?e  and  cosine  of  the  difference  of  two  angles. 
Let  the  angle  between  y  and  a  (Fig.  43)  be  vj/.     Then 

7       a  °y' 


GEOMETRIC    MULTIPLICATION    AND    DIVISION.  87 

in  which 

-  =  cos(i/^  -0)-  esin(i/'  -  6) , 

B 

-  =  cos6'-fcsiii^. 
a 

"  =  cost/^  —  esiui//, 
7 

and.  as  in  the  preceding  example, 

cos(i/'  —  $)  =  cosO  cosi/'  -}-sin^  sin  i//, 
sin  {\l/  —  6)  —  cosO  sin  i/^  —  sin^  cost/'. 

11.  //'  «  straifjht  line  intersect  two  other  straifjld  lines  so  as  to 
make  the  alternate  angles  eqnaU  the  tico  lines  are  jxtrallel. 

Let  a  and  y  (Fig.  44)  lie  nnit  vectors  along  ab  and  cd,  and 
B  a  unit  vector  along  ac     Then 


a/3  =  —  cos^  +  esin^, 

Fi-r 

.  44. 

fSy  —  —  cosO  —  esinO  ■ 

y  =  a. 

B           «. 

V 

whence 

/3 

< 

and  therefore,  liquation  (•"><))' 

/ 

If  a  =  abJ  tlien 

al3  =  < 

cos^  — 

£sin^, 

/3y  =  - 

-cos^ 

—  £sin^. 

—  a. 

[Eq.  (.^;y 

12.  If  a  jiarallelor/ram  he  described  on  the  diaijonals  of  av>/ 
parallelor/ram,  the  area  of  the  former  is  twice  that  of  the 
latter. 

Let  a  and  /3  represent  the  sides  as  vectors  ;  then  the  diagonals 
are  a  +  /?  and  u  —  /?,  and 

V(a  +  (3)  (a  -  l3)=\ifSa  -  afS)  =  2  V/iu. 

since  Va^  =  V/3-  =  0  and  -Va/3  =  y(3a. 


88  QUATEENIONS. 

But,  from  the  order  of  the  factors, 

UV(a  +  /5)(a-/S)=Uy^a, 
hence 

TV  (a  +  ft)  {a- (3)^2  TV^a, 

which  is  the  proposition  (Art.  41,  7). 

13.  Parullelo(jrams  on  the  same  base  and  hetiveen  the  same 
2KiraUeIs  are  equal. 

We  have  (Fig.  45) 

Fig.  45. 

BE  =  BA  +  AE 
=  BA  +  XHC. 

Operating  with  V .  bc  x 

V(bc  .  be)=V(bc  .  ba), 
since  V.rBC-  =  0. 
Hence 

BC  .  BE  sin  EBC  =  BC   .  BA  sin  AHC, 

which  is  also  true  when  tlie  bases  are  equal,  but  not  co-incident. 

14.  If ,  from  any  x)oint  in  the  2^l((ne  of  a  parallelogram.,  j)^^'- 

pendicidars  are  let  fall  on  the  diag- 
onal and  the  two  sides  that  contain 
it,  the  jrroduct  of  the  diagonal  and 
its  perpendicular  is  equal  to  the 
Slim,  or  difference,  of  the  products 
of  the  sides  and  their  o'esjjective  per- 
2)endicidars,  as  the  j)oint  lies  with- 
out or  ivithin  the  ixirallelogram. 

Let  OA  =  a,  OB  =  /?,  OP  =  p  (Fig.  46) . 
Then 

Vap+V/3p=V(a  +  /3)p. 

But 

UVap  =  UV/3p=  UV(a  +  ft)p. 


Hence 


TVap  +  TV/5/J  =  TV(a  +  ft)  p. 


GEOMETEIC    MULTIPLICATION   AND    DIVISION. 


80 


For  p'  =  or',  we  hjvvo 

V\ap'=  -  Vy(3p'=  ±  UV(a  +  f3)p'', 
.-.    T\ap'-T\(3p'=       'I\{a+(3)p: 

15.  7/",  o?i  a?*?/  two  sides  of  a  triangle^  as  ac,  ar  (Fig.  47), 
awj  two  exterior  parallelograms,  as  acfg,  adeb,  be  constructed, 
and  the  sides  ed,  gf,  produced  to  meet  in  ii,  then  will  the  stim  of 
the  areas  of  the  parcdlchgrams  be  equal  to  that  tvhose  sides  are 
equal  and  jiarallel  to  cb  and  ah. 

Lot    AE  =  a,    AB  =  /?,    AC  =  y 
and  AG  =  8.     Then 


AH  =  AE  +  EH 
=  a  -  Xp. 

Operating  with  x  V  .  y8 

Y(AH.;8)=Va^.         {a) 

We  have  also 


Operating  with  x  V  .  y 


AH  =  AG  +  GH 

=  8  -  7/y. 


V(An  .  y)=VSy. 
Hence,  from  («)  and  (&), 

VAH(/8-y)=Va^-V8y, 

Y(ah  .  cb)  =  Va^  -VSy  =  Va^  +Vy8. 


(6) 


These  vectors  have  a  common  versor ;  whence  the  proposition. 
If  one  of  the  parallelograms,  as  ad',  be  interior,  then  ae'=  —  a 
and  ah'  =  —  a  —  .tjS  =  8  +  »/y,  and 

V(AH'.;8)  =  -Va^, 

V(An'.  y)=V8y; 

.-.      Vah'(^  -  y)  =  -Va/3  -  V8y  =  V/8a  -  V8y. 


90  QUATEENIONS. 

But  in  this  case 

UV(An' .  cb)  =  -  UVySa  =  -  UYSy, 

and  tlie  area  of  the  parallelogram  on  aii',  cb,  is  the  area  of  af 
minus  the  area  of  ad'. 

IG.    To  find  the  angle  behveen  the  diagonals  of  a  parallelogram. 

Let    AD=:BC  =  a    (Fig.    48), 

Fig.  48.  and  BA  =  CD  =  /3,  d  and  d'  being 

A^  J^  r,     the    tensors    of    the   diagonals. 

Then 

AC  .  DB  =  -  (a  -  ^)  (a  +  /3) 
=  -a--2Va/3-|-yS^ 

Taking  the  sealars 

cos  DOC  .  dd'=  a-  —  W. 
Taking  the  vectors 

sin  DOC  .  dd'=2ahii\xiO, 
since  UV(ac  .  db)  =  —  UVa/3. 

.'.  tan  DOC  =  —  tan. 


2a7^sin^ 


cr  -  b- 

17.  The  sum  of  the  sqtiares  on  the  diagonals  of  a  parallelo- 
gram equals  the  sum  of  the  squares  on  the  sides. 

In  Fig.  48 

BD-  =  (a  +  /S)  '  =  a'  +  2  SafS  +/S', 
CA-  =  (/3  -  a)-  =  ^-  -  2  Sa/?  +  a-  ; 
.-.    CA-  +  BD-=2a-  +  2^-, 

or 

BD-  +  CA-  =  BA-  +  AD-  +  DC"  +  CB". 

18.  The  sum  of  the  squares  of  the  diagonals  of  any  quadri- 
lateral is  twice  the  sum  of  the  squares  of  the  lines  joining  the 
middle  2'>oints  of  the  o^yposite  sides. 


GEOMETRIC    MULTIPLICATION   AND    DIVISION. 


01 


Fig. 

49. 

I>_^ 

V. 

~r~~ 

__;/ 

o^ 

^f^ 

/ 

/ 

L 

Let  AK  =  a,  Ai)  =  /8,  i)C  =  y  (Fig.  49).  For  the  squares  of 
the  diagonals,  we  have 

(^  +  y)•^  +  (^-a)^ 

and  for  the  bisecting  lines 

Whence  the  proi)osition  readily  follows. 

19.  The  sum  of  the  squares  of  the  sides  of  any  quadnlateral 
exceeds  the  sum  of  the  squares  on  the  diagonals  by  four  times  the 
square  of  the  line  joining  the  middle  points  of  the  diagonals. 

Let    Ali  =  a,     AC  =  y8,     AD  =  y 

(Fig.   .")0).      The  squares  of  the  Fig.  so. 

sides  as  vectors  are  o 

a^  +  (^-a)^  +  (y-/3)^  +  yS 

or 

2(a-  +  /32  +  y2)-2S^a-2Sy/3. 

The  squares  of  the  diagonals  are 

/3^  +  (y-a)^ 

or 

;S=  +  y-  +  a--2Sya. 

The  former  sum  exceeds  the  latter  h\ 

a-  -f-  ^'  -I-  r  -  ■-  ^1^"-  -  -  ^y/5  +  -  Sya 

or  bv 

which  may  be  put  under  the  form 


92 


QUATERNIONS. 


But  °~-^  =  AO,  and   —  ~  =  sa.     Substituting  these  values, 

we  obtain 

4(ao  +  sa)-,  or  4so^, 

which  is  also  true  of  the  vector  lengths. 

20.    Ill  any  quadrilateral,  if  the  lines  joining  the  middle  points 
of  opposite  sides  are  at  right  angles,  the  diagonals  are  equal. 
A7ith  the  notation  of  Fig.  49,  we  have 

FE.<ai=[^(y-a)  +  ^]i(a+y). 

But,  bv  condition. 

S(.E.on)  =  r_«V^  +  ^  =  0. 
^  ^44^2^2 


Whence 


(y  +  ^)^  =  (^-a)^ 


AC-  =  BD^, 
AC  =  BD. 


21,  In  any  quadrilateral  prism,  the 
sum  of  the  squares  of  the  edges  exceeds 
the  sum,  of  the  squares  of  the  diagonals 
by  eight  times  the  square  of  the  line 
joining  the  points  of  intersection  of  the 
tivo  pairs  of  diagonals. 

Let  OA  =  a,  oB  =  /3,  oc  =  y,  CD  =  8 
(Fig.  ol).  For  the  sum  of  the 
squares  of  the  edges  we  have 


2[2a-  +  2/5-  +  2y2  +  28-'  -  2.S8a  -  2S3^].  (a) 

The  sum  of  the  squares  of  the  diagonals  is 

(y  +  S)'  +  (y-8)'  +  (y  +  a-/3)2-f-(y  +  ^-a)S 

2(a-+/S-  +  8-  +  2y--2Sa/?).  {b) 


GEOMETRIC    IMULTITLICAIION    AND    DIVISION.  93 

Tho  vectors  to  the  intorscetious  of  the  duigonals  are 
1(8  +  7)     and     .l(y  +  a  +  /3), 
and  the  vector  joining  these  points  is 

i(a  +  ;3-6). 

Squaring  and  multiplying  l>v  eiglit,  we  have 

2  [a-  +  ;8-  +  S-  +  2  Sa/^  -  2  Su8  -  2  S)8S] , 
whicli  added  to  {b)  gives  (a). 

22.  /h  a»y  tetmedron,  if  tiro  pairs  of  ojiposite  edges  are  at 
right  angles,  respectively,  the  third  pair  iviU  he  at  right  angles. 

Let  OA  =  a,  OB  =  /?,  oc  =  y  (Fig.  52) . 

The  conditions  give  '*^; 

Sa(/3-y)=:0,  A\ 

S/3(a-y)  =  0.  //       \ 

Subtracting  the  first  of  these  equa-           /        /  \ 

tions  from  the  second  o<^- / -—^b 

S7(a-/3)  =  0,  nI/-^^^ 

A 

which  is  the  proposition. 

23.  To  find  the  relations, between  the  edges,  angles  and  plane 
areas  of  a  tetraedron. 

With  the  notation  of  Fig.  .'i2,  we  liave 

CA  .  CB  =  {a-y)((3-y), 

or 

CA  .  CB  =  a/^  — ay  —  y/3  +  7  .  («) 

Representing  the  tensors  of  oa  and  cb  hy  m  and  n,  and  taking 
the  scalars  of  («), 

S(CA  .  cb)  =  Sa/3  —  Say  —  Sy/3  +  7", 

whence 

(2  _  (<c  cos  Aoc  —  be  cos  Boc  =  mn  cos  acb  —  ab  cos  aob, 


94 


QUATERNIONS. 


wliicli  is  the  relation   between    the    edges    and   their   Inchided 
angles. 

Taking  the  vectors  of  (a),  and  sqnaring, 


[V(CA  .  Cb)]-  =  (Va/5)-  -  Va^Vay  -  Ya^Yy^  -  YayYayS  ) 

+  (Yay)^  +YayYyyS  - Yy^Ya^  +Yy/5Yay  +  (Yy/?)".  I 


(b) 


But 


i\a(3Yy/3  +  Yy/3Ya/S)  =  -  2  S  .  Ya^Yy^       (Eq.  55) 
=  2T\a/3T\y/3  COSB, 


in  which  b  is  the  angle  between  the  planes  aob,  boc. 
Also 

-  (Ya/5Yay  +YayYa^)  =  -  2  S  .  Ya^Yay  =  2TYa;STYay  COS  A, 

and 

YayYy;8  +  Yy/3Yay  =  2  S  .  YayYy/?  =  -  2  TYayTYy^COS  (180°-  c) 

=.2TYayTYy/5cosc, 

in  which  a,  b  and  c  are  the  angles 
opposite  the  edges  bc,  ac  and  ab  re- 
spectively.    Hence  (b)  becomes 

-  [TY(CA  .  cb)]^=  -  (TYa/S)--  (TYay)- 

-(TYy/5)^ 
+  2  T Ya^  T Yay  COS  A  +  2  T Ya/5TYy/3  COS  B 
+  2TYayTYy;8cosc. 

But  (Art.  41,  7th) 

TY(cA  .  cb)  =  2  area  acb, 

and  similarly  for  the  others.     Hence,  dividing  by  —4, 

(area ABC)-=  (area  aob)-  +  (area  aoc)-  +  (area  boc)-  — 

2  area  aob  area  aoc  cos  a  —  2  area  aob  area  boc  cos  b  — 
2  area  aoc  area  boc  cos  c  , 

which  is  the  relation  between  the  plane  faces  and  their  included 
ano-les. 


GEOMETKir    ]SirLTIPLICATIOX    AM)    DIVISION. 


05 


Utile  niinU's  arc  ri-iht  aniih's.  tlicii 

(area  abc)-'  =  (aiva  xov.)-  +  (area  aoc)-  +  (area  uoc)-. 

24.    To  inscribe  a  ciirJc  in.  a  (jivcn  iriany/c 

Let  a,  fS,  y  (Fig.  o3)  be   unit  vec- 
tors along  the  sides.     Then,  Art.  IG,  '^' 
the  angle-bisectors  are 


.^■(/5  +  y), 
.'/(y  +  "), 
z{a-/3). 


Now 


a:(f3  +  y)  =  cy->/{y  +  a). 

Operating  with  Y  .  (y  +  a)  x 

X 

Hence 


ay 


Vy/J-fVa/3-fVay 
Vay 


or,  since  u,  ^,  y  are  unit  vectors, 


(/^  +  y), 


sin  A  -f  siun  +  suic 


(^  +  y)- 


Squaring,  to  find  the  length  of  ao,  we  have,  since  {(3-{-y)- 
■2(1  + cos  a), 

'-AO^=-r '•-'""  .        T2(l-fCOSA), 

[sin  A  +snn!  +  snicj 

-— V-2(l  +  COSA), 
COSti^A. 


sin  A  +  siuB  +  snic 
r  sin  B 


sin  A  +  sui  r.  +  snic 


25.  If  tangents  he  dniivn  at  the  vertices  of  a  triangle  inscrihed 
in  a  circle,  their  intersections  loith  the  opposite  sides  of  the  triangle 
will  lie  in  a  straight  line. 


96  QUATERNIONS. 

Let  o  be  the  center  of  the  circle  (Fig.  54)  whose  radius  is  r, 
and  OA  =  a,  OB  =  ^,  oc  =  y.  Since  oa  and  ap  are  at  right 
angles, 

S(OA  .  AP)  ==  0. 

But 

AP  =  AB  +  BP  =  Ali  +  ?/BC  =  y8  —  a  +  ?/(y  —  ^)  ; 

^'^'  ^-  hence,  substituting  this  value  above, 

Sa[^-a  +  7/(y-^)]=0, 
.,,_  r  +  fia/3    . 


Say  —  Sa/3 


Nay  —  ha/3 


_  (r  +  Say)^  -  (r-  +  Sa;8)Y 
Say  -  Sa/3  " 

Similarly,  or,  ))y  a  cyclic  chano-e  of  vectors. 


0(.  =  (!l  +  j<:^/?)Y-(>-^  +  S;8Y)a 
Sa;8  -  S^y 

^p  _  (r  +  S^y)»  -  (/•-  +  Say)B 
S/3y  —  Say 


Whenc( 


(Say  -  Sdi,/3)op  +  (Sx^  -  S^y)OQ  +  {S(3y  -  Say)oU  =  0. 

But  also 

(Say  -  S.^)  +  (Sa/3  -  S,9y)  +  (N^y  -  Say)  =  0. 

Hence  p,  q  and  u  are  coUinear. 

2G.    The  sum  of  the  cuujles  of  a  triangle  is  two  right  angles. 


GEOMETRIC    MULTIPI.ICATION    AND    HIVISION. 


Lot   a, 

Then  (Art.  42) 


1)0  unit  voctors  iilong  I'.c  ex  luul  au  (Fig.  •J'>)- 


a         i^ 

-  =  e  "  , 

y 

—  =  fc       1 

a 

20 


But 


-    1 
7    ^ 


■y  ^  -4      1(0  +  0  +  ^) 


Hence  l(cf> +  6 +  ip)  =  an  even  multiple  of  2  (Art.  42) ,  as  2  », 

as  we  go  round  the  triangle  n  times. 

In  taking  the  arithmetical  sum,  or  passing  once  round,  we 
take  the  first  even  nuiltiple  of  2,  or 


£(.^+^ +  >;/)  =  4 


2  TT  =  -,  or  two  right 


and  the  sum  of  the  interior  angles  is  : 
angles. 

27.     The  angles  at  the  base  of  an  isosceles  triangle  are  equal  to 
each  other. 

Let  a  and  ft  (Fig.  •>(<)  be  the  vector  sides 
of  the  triangle,  and  Ta  =  T(3.  Then,  if  the 
proposition  be  true, 

n-13  fS-.' 

or 

a(a-^)-'=K/5(/3-a)-i=(/3-a)-'/?, 

a(^-a)  =  (a-/3)/?; 


which  is  true,  since  Ta  =  T/?. 


98 


QUATEENIONS. 


28.  To  find  a  point  on  the  base  of  a  triangle  such  that,  if  lines 
be  drawn  through  it  jxiraUel  to  and  limited  by  the  sides,  they  will 
be  equal. 

Fig- 5"-  Draw  DE  (Fig.  57)  and  df  parallel  to 

the    sides.     From    similar   triangles,   if 
AE  =  ;i-AC, 

AE        BF        BA  — j^F 


whence 


Now 


AC        KA 


1-.1- 


or,  smce  fd  =  ae, 


AD  =  AF  +  FD, 

(1  — a')AB  +a-AC. 
But,  since  fd  is  to  l)e  equal  to  ed, 

( 1  —  .r)  Tab  =  a-TAC  =  ?/ ; 
.-.    (1  — .i')TabUab  =  ?/Uab, 
x-TacUac  =  ^Uac, 

and  therefore 

ad  =  »/(Uab  +  Uac)  , 

and  D  is  on  the  angle-hiseetor. 

29.    Jf  any  line  he  drawn  through  the  middle  point  of  a  line 

joining  two  parallels^  it  is  bisected  at  that 

58-  point. 

II 

30.  In    any  right-angled    triangle  abc 

,j^     (Fig.  58),  the  lines  bk,  cf,  al  meet  in  a 
p)oint. 

31.  In  any  triangle  the  sinn  of  the 
squares  of  the  lines  i,u.  kk,  df  (Fig.  5.S) 
/,s  three  times  the  sum  of  the  squares  of  the 
sides  of  the  trianglr. 

32.     The  sum  of  the  angles  about  two  right  lines  which  intersect 
is  four  right  angles. 


i    UNIVER 


(GEOMETRIC    ]MULTrPLICATION   AND   DIVISION.  W 

33.  If  the  sides  of  an>/  poh/f/on  he  prodnred  so  as  lo  form 
one  angle  at  each  vertex,  the  sum  of  the  (UKjJes  is  four  rit/U 
angles. 

34.  Find  the  eight  roots  of  nnitij  (Art.  31)). 

3.').  The  square  of  the  medial  to  ainj  side  of  a  triangle  is  oue- 
half  the  sum  of  the  squares  of  the  sides  tchich  contain  it.,  minus 
one-fourth  the  square  of  the  third  side. 

55.    Product  of  t-wo  or  raore  Vectors. 

1 .     Let  q  —  n/3,  r  =  y.      'riicii,  since  Sr/y  =  S/vy, 
Su/iy  =  SyayS. 

Let  q  =  ya,  /•  =  /j.     Then 

Sqr  =  Srq  =  HyajS  =  HfSya  ; 
.-.    Sa^y  =  S/Sya  =  Sya/? (lOS), 

or,  the  scfdar  of  the  product  of  three  vectors  is  the  s((7ne  if  the 
cj/clical  order  is  not  changed. 

This  may  also  he  shown  In*  means  of  the  associative  hiw  of 
vector  innltii)hcation  as  follows  : 

a,Gy  =  (a^)y  =  (S,.^+Va/?)y. 

Taking"  the  scalars 

Sa/5y=S(.Sa/?+VayG)y 

=  S(Va^  .  y),    since    S(Sa/?  .  y)  =  0, 
=  S  .  yVa^  ; 

introducing-  the  term  S  .  yHafS  =  0, 

=  S  .  y VayS  +  S  .  ySa/5 
=:S.y(Sa/?+Va^) 
=  Sy(a^)  =  Sya/?. 


100  QUATERNIONS. 

In  a  similar  manner 

Sa;8y=S  .  a(S/?y+V^y) 
=  S  .  aV/3y 
=  S(Yy8y  .  a) 
=  S(Yy8y  +  S^y)a 
=  S/5ya, 


and,  as  before, 
2.  Again 


Sa/3y  =  S;Sya  =  Sya^. 

Sa^y=S.a(S/3y+V;8y) 
=  S  .  aVySy 
=  -  S  .  aYy/3 
=  -Sa(Yy^  +  Sy/3); 
.-.    Sa^y  =  -Say/? (109), 

or,  a  change  in  the  cyclical  order  of  three  vectors  changes  the  sig)t, 
of  the  scalar  of  their  product. 

3.  Resuming 

a^y  =  a(^y) 

and  taking  the  vectors, 

Va/3y=V.  a(S,8y+V/3y) 
=  aS/3y  +V  .  aYySy. 

Vy/?a=V(Sy^+Vy^)a 

=  V.  aSy/3-y.  aVy^ 
=  V.  aSy/?+V.  aV/3y 
=  V.a(Sy;8  +  V^y) 
=  aSySy  +  V  .  aVySy  ; 
.-.    Va/3y  =  Vy/3a (110), 

or,  the  vector  of  the  product  of  three  vectors  is  the  same  as  the 
vector  of  their  product  in  inverted  order. 

4.  Geometrical  inteipretation  of  Sa^y. 

T>et  o,  /?,  y  be  unit  vectors  along  the  three  adjacent  edges  oa, 
oj!,  oc  (Fig.  59)  of  an}'  parallelopiped,  0  being  the  angle  be- 


Also 


GEOMETRIC    MULTIPLICATION    AND    DIVISION.        101 

twecn  a  and  /3,  and  0'  the  an^ie  made  by  y  with  the  plane  aoh. 
Then 

a/?  =  —  cos^  4-  €  sin^, 

€  being  a  A-ector  peri);Midit'ular  to  tlie  phme  aou. 
Operating  with  X  S  .  y 

SaySy  =  S(  -  cos (9  +  e  sin^)y  Fig.  50. 

=  S(sin^  .  cy). 


Bnt   Ssy  =  —  cos   of   the    angle 
between  e  and  y  =  —  sin^' ; 

.*.  Sa/3y  =  —  sin^/sin^; 

Now,  if  a,  ^,  y  represent  as  vectors  the  edges  oa,  ob,  oc. 
whose  lengths  are  «,  ?>,  c, 

SaySy  =  -  TaT/>'Ty  sin  ^  sin  6' 
=  —  a?^csin^  sin^' 

But  a&sin^  =  area  of  tlie  parallelogram  whose  sides  are  a  and 
h,  and  csin6''  =  perpendicular  from  c  on  the  plane  aou.     Hence 

—  Sa/3y  =  volume  of  a  X)araUi'lopiped  irhose  edges  are 
a,  h  (Old  c,  drawn  parallel  to  a,  /3  and  y. 

Cor.  .1.  Whatever  the  order  of  the  vectors,  the  volume  is  the 
same  ;  hence,  as  already  shown, 

±  Sa^y  =  ±  Hf^ya  =  ±  Sya/?  =  T  Say^,  etc. 

Cor.  2.  If  SaySy  =  0,  neiUier  a,  /3  or  y  being  zero,  then  cither 
^  =  0,  or  $'  =  0,  or  the  vectors  are  complanar. 

Cor.  3.    Conversely,  if  a,  /?,  y  are  complanar,  Sa/5y  =  0. 

Cor.  4.  The  volume  of  the  triangular  pyramid  of  wliiili  the 
edges  are  oc,  ob,  oa,  is  —  ^  SajSy. 

5.  AVe  have  seen  that  when  a,  /3  and  y  are  complanar,  SaySy=0, 
and  therefore  a/3y  is  a  vector.     To  lind  this  vector,  suppose  a 


102  QUATERNIONS. 

triangle  constvueted  whose  sides  ab,  bc,  ca  have  the  directions 
of  a,  /?  and  7  respective!}',  a  vector  not  being  changed  b}-  motion 
parallel  to  itself.  Since  the  tensor  of  the  vector  sought  is  the  prod- 
uct of  the  tensors  of  a,  ^  and  y,  we  have  to  find  U(ab  .  b<;  .  ca)  , 
?".e.,  its  direction.  Circumscribe  on  the  triangle  abc  a  circle  and 
draw  a  tangent  at  a,  represented  b}-  t'at.  Since  the  angles  tab 
and  bca  are  equal,  we  have 


ca  at  |_ 


U(bc  .  CA)  =  U(ab  .  at')  [=  U(ba  .  at)]. 
Introducing  Uab  X 

U(aB  .  bc  .  CA)  =  U(aB  .  AB  .  At')[  =  U(aB  .  BA  .  At)  ]  , 

or,  since       TJ(ab  ,  ba)  =  —  (U  .  ab)-=1, 

U  ( AB  .  BC  .  ca  )  =  —  U  .  at'  =  U  .  AT. 

Hence,  if  a,  b,  c  are  any  three  non-collinear  points  in  a  plane, 
or  if  a,  /?,  y  are  the  sides  of  a  triangle  joining  them,  in  order 
(in  either  direction,  since  Va/3y  =  Yy/3a) , 

a^y,      ^ya,      ya/3 

are  the  vector  tangents  to  the  circumscribing  circle  at  the  angles 
of  the  triangle. 

Again,  if  a,  b,  c  are  any  three  points  in  a  plane,  not  in  a 
straight  line,  and  a  and  /?  are  two  vectors  along  the  two  succes- 
sive sides  AB,  BC  of  the  triangle  which  the}-  determine,  and  cd  a 
vector  drawn  from  c  parallel  to  y,  intersecting  the  circumscribed 
circle  at  d,  then  is  da  parallel  to  Ya/3y  =  8.     For 

8  =  a/3y  =  a^f^y  =  af^'/S'^y  =  -  (T^)^«/^-V  -  -  (T^)^'^y, 

whence  U  .  — ^,  which  turns  /3  parallel  to  —  a,  turns  y  into  a 

direction  8  =  da',  the  opposite  angles  of  an  inscribed  (luadrilateral 
being  supplementary. 


GEOMETRIC    MULTIPLICATION    AND    DIVISION.        103 

If  y  luivo  11  direction   such  that  CD  crosses  Ai?,  or  the  (ju:uh-i- 

lateral  is  a  crossed  cue,  it  is  evident  on   construction   ol"  tlic 

figure  that 

U8'=  U«^y  =  U(ad)  =  -  U8. 

Hence  the  continued  product  of  the  three  successive  vectoi- 

sides  of  a  quadrihiteral  inscribed  in  a  circle  is  parallel  to  tiu- 

fourth  side,  its  direction  being  towards  or  from  the  initial  point 

as  the  quadrilateral  is  uncrossed  or  crossed  ;  and,  conversely,  no 

plane  quadrilateral  can  satisf}'  the  above  formula  ±  US  =  Ua^y. 

unless  A,  15,  c  and  d  are  eon-circular.     The  continued  product 

of  the  four  successive  sides  of  an  inscribed  (luadrilateral  is  a 

scalar,  for 

a/3y8  =  (a/3y)  B=±8^  =  ^d-.    ■ 

Since  tlie  product  of  two  vectors  is  a  quaternion  whose  axis  is 
perpendicular  to  their  plane,  while  the  product  of  a  quaternion 
b}'  a  vector  perpendicular  to  its  axis  is  another  vector  perpen- 
dicular to  its  axis,  and  so  on,  it  follows  that  the  continued 
product  of  an}'  even  number  of  complanar  vectors  is  generally  a 
quaternion  whose  axis  is  perpendicular  to  their  plane,  while  the 
product  of  any  odd  number  of  complanar  vectors  is  a  A'ector  in 
the  same  plane.     Hence  the  formulae 

Sa=0.       Sa/?y=0,       Sa/3ySo- =  0,       etC, 

for  complanar  vectors. 

If,  however,  the  given  vectors  are  parallel  to  the  sides  of  a 
polygon  ABC MX  inscribed  in  a  circle,  then 


U(ab  .  BC  .  CD MX  .  xa)=  r(Ar.  .  r.c  .  ca)  r(vc  .  CD  .  da)  

X  U(am  .  MX  .  xa). 

But  each  of  the  products  U(ab  .  bc  .  ca)  is  equal  to  U  .  at. 
AT  being  the  tangent  to  the  circle  at  a.     Hence 

U(aB   ,  BC  .  CD  MX  .  xa)  =  (U  .  at)". 

which  reduces,  according  as  n  is  even  or  odd.  to  ±  1  or  ±1'  .  at. 
Hence  the  product  of  the  vectors  will  be  a  scalar  or  a  vector 


104  QUATERNIONS. 

according  as  thoir  number  is  even  or  odd,  and  in  the  latter  ease 
this  vector  is  parallel  to  the  tangent  at  a. 

If  the  vectors  are  not  coniplanar,  but  parallel  to  the  successive 
sides  of  a  gauche  pol^'gon  inscribed  in  a  sphere,  the  polygon 
may  be  divided  as  above  into  triangles,  for  each  of  which  the 
product  of  the  three  successive  sides  is  a  vector  tangent  to  the 
circumscribing  circle,  all  these  vectors  l^ing  in  the  tangent  plane 
to  the  sphere  at  the  initial  point.  If  the  number  of  sides  is  even, 
their  product  will  be  a  quaternion  whose  axis  is  perpendicular  to 
the  tangent  plane,  i.e.,  lies  in  the  direction  of  the  radius  of  the 
sphere  to  the  initial  point ;  if  odd,  the  product  is  a  vector  in  the 
tangent  plane. 

Hence,  if  a,  b,  c  and  d  are  four  given  points,  not  in  a  plane, 
AB  =  a,  BC  =  /?,  CD  =  y  being  given  vectors,  and  p  any  other 
point  such  that  dp  =  o-,  pa  =  p,  if  p  lies  on  the  surface  of  a 
sphere  through  the  four  given  points,  we  have  the  necessary  and 
sufficient  condition 

a^ycrp  =  pa-yfSa, 

for  each  member  is  equal  to  minus  the  conjugate  of  the  other, 
and  must  therefore  (Art.  4G)  be  a  vector. 

6.    From  Equation  (56), 

^y-y^=2V^y. 

Operating  with  Y .  a  x 

2Y.ay/3y=T.a(^y-y^). 

Introducing  in  the  second  member  /3ay  —  /Say, 

=  V  (a^y  -  ay/3  +  ^ay  -  /Say) 
-  V(a/5  +  ;8a)y  -  V(ay^  +  ya/3) 
=  V»2(Sa^)y-V(ay+ya)/3 
=  2ySa^-2^Say. 

Hence 

y.a\(3y  =  ySxl3-/3Hiy       .      .      .      .      (111). 


GEOMETRIC    MULTIPLICATION    AND    DIVISION.        10.") 

This  formula  inuv  1h'  oxUmuUhI.     Tims,  for  a  write   Va8.  ami 
we  have  ^,  ^  ^.^^^^.^^  ^  yS(VaS);8  -  ^S(VaS)y, 

V  .  V<i8V/3y  =  ySa8^  -  /3SaSy  .      .      .      .      (112). 

An  inspection  of  this  fornuiUi  shows  that   it  gives  a  vector 
complanar  with  y  and  f3.     INIoreover,  since 

Y  .  Va8y/3y  =  V  .  Vy/8VaS  =  8Sy/8tt  -  aSy/8S, 

it  is  also  complanar  with  a  and  S,  and  is,  therefore.  i)arallel  to 
the  line  of  intersection  of  the  planes  of  a,  8,  and  /3,  y. 
Similarly 

V  .  V/3y YaS  =  8S,8ya  -  aS^yS  =  -  Y  .  Ya8Y/3y   .      (113). 

Adding  Equations  (112)  and  (113) 

8S;8ytt  -  aS/?y8  +  ySaS/3  - /8Sa8y  =  0     .      .      (114), 
or 

8Sa|8y  =  aS^y8  - /3Say8  +  ySa^8        .       .       .      (115), 

a  formula  expressing  a  vector  8  in  terms  of  anj'  three  given  di- 
planar  vectors,  a,  /8,  y  ;  so  that,  if 

S/3y8  =  /v,       -  Say8  =  SyaS  =  C,      Sa/S8  =  a,      Sa/3y  =  m, 
8=  m-\ba  +  c/3-\-ay). 

7.    Resuming  Equation  (111),  and  adding  aS/3y  to  both  mem- 

^^^'^'  Y  .  aY^Sy  +  aS/3y  =  ySa/3  -  /SSay  +  aS/?y, 

whence 

Y.a(S^y  +  Vi8y)  = 

Ya^y  =  aS;8y  - /8Say  +  ySa^  .      .      .      .      (HC). 

The  form  of  this  equation  show\s  that  a  and  y  may  be  inter- 
changed, or  that  \a(3y  =  Yy/3a,  as  already  shown. 
Again,  replacing  a  by  Y«/S  in  Equation  (111), 

Y  .  Ya^Y/8y  =  yS(Ya/3)/S  -  /3S(  Y«/?)y, 
or 

y.Ya(3\f3y  =  -fSSa/3y (117). 


106  QUATERNIONS. 

8.    Writing  Yy^a/B  first  as  V(y  .  Sa/3) ,  and  then  as  T(y8  .  a/?), 

we  have 

y(y  .  Sa/3)  =  Y  .  y{SSa j3  +  VSa/3) 

=  ySSa^  +V  .  yJSafS 
[Equation   (11 G)]       =  ySa^8 +Yy8Sa;S -VyaS3/5 +Vy/?S3a.      (a) 
V(y8  .  «/3)  =V(SyS  +Vy8)  (Sa/3  +Va/3) 

=  VySSayS  +Va^Sy8  +V  .  Vy8Vu;8 

=  Vy8Sa;8  +Ta^Sy8  -V  .  Vay8Vy8, 
or,  Equation  (112), 

=  Vy8Sa/3  +Vay8Sy8  -  8Sa/Sy+  ySa/3S.        (6) 

Equating  (a)  and  (6), 

8Say8y  =  V;8ySa8 +VyaS)8S +Vay8Sy8     .      .      (118), 

a  formula  expressing  a  vector  8  in  terms  of  tliree  otlier  vectors 
resulting  from  their  products  taken  two  and  two  ;  so  tliat,  if 
Sa^y  =  m,  Sa8  =  0,  S^8  =  b,  SyS  =  c, 

8  =  Hi-1  (aV^y  +  hYya  +  cVayS) . 
Operating  on  Equation  (118)  with  S  .  p  X  ,  we  obtain,  since 

S  .  pYya  =  Spya , 

So8Sa/3y  -  S/38Spya  -  Sa8Sp/8y  -  Sy8S/)a/3  =  0, 
^^  Sa8Sp/3y  -  S/3SSypa  +  SySSpayS  -  SpSSa/3y  =  0  .      (119), 

a  formula  eliminating  8. 

56.   Exercises. 

Prove  the  following  relations  : 

1.  Sa^yS  =  SSa/3y. 

2.  a/3  .  ^y  =  —  ay. 

3.  a2/3-=a/3  .^a. 

4.  S  .  Va/8V/3y  =  S  .  a/SV/?y (120). 

5.  Sa^y8  =  Sa/3Sy8-SayS/38  +  Sa8S/3/ (121), 

from  wliieh  show  that  Su/3y8  =  SfSy^a. 


GEOMETRIC    MULTIPLICATION    AND    DIVISION.        107 

f).  S  .  Vaj8Vy8  =  Sa8Sy8y  -  SayS;88 (122). 

7.  S(«+^)(/?  +  y)(y  +  a)  =  2Say8y. 

8.  a/3y  +  yySa  =  2  Va/3y. 
I».  n/?y  -  y;8a  =  2  Saygy. 

10.  y(aV/3y+/8Vya  +  yVa/S)=0 (123). 

1  1 .  Vaygy  -I-  Vya/3  =  2  ySa(3. 

1 2 .  aV/?y  +  y8  Vytt  +  yVa/S  =  3  Sa/3y. 

13.  S  .  VaySVySyVya  =  -  (Sa^y)". 

14.  S  .  y V/Sa  =  y/?a  -  yS^a  +  ^Sya  -  aS/Sy         .       .       .      ( 1  24  ) . 

15.  S  .  J{\a/3\fty)  V(V^yVya)  V(VyaV«/3)  =  -  (Su;8y)^ 

IG.    S  [Va;8Vy8  +  VayVS/?  +  VaSV^y]  =  0       .      .      .      .      (125). 

17.  If  Saygy  =  ?)i,  Sap  =  0,  S;8p  =  0,  Syp  =  0,  show  that  p  =  0. 

•  Conversel}',  if  p  is  uot  zero,  then  Sa^Sy  =  0. 

18.  Interpret  p  =  a" '/3a. 

AVe  have  first,  directlv, 

Tp  =  T^, 
Sap/3  =  Saa-'/3a/3  =  SySayS  =  S^'a  =  0  ; 

.'.  p,  a  and  j3  are  complanar. 

Sap  =  Saa~'y3a  =  SySa, 

—  TpTa  COS  0  =  -  TaTyS  COS  <^, 

or.  since  Tp  =  T/3,  cos^  =  cos  (f>. 

Similarly  Yap  —  Y/Ja,  and  nin^  =  sin^.     Hence 

and  a  bisects  the  angle  between  /3  and  p. 

19.  Show  that  p  =  a/3a-i  =  a-\iia(3  -  Vayg) . 

20.  p  being  any  vector,  show  that  V  .  YapYp^g  =  .Tp, 

21.  If  SriyS  =  —a-,  show  that  a  is  perpendicular  to  /J  —  a. 

22.  What  are  the  relative  directions  of  a  and  /3,  if  K-  = ? 

B      B  '^  a  « 

If  k"  =  "? 


108 


QUATERNIONS. 


57.   Examples. 

1.    The  altitudes  of  a  triangle  intersect  in  a  point. 

Let  (Fig.  60)  AC  =  /3,   cb  =  a,   ah  =  y. 
Then  vectors  along  c'c,  f/b  and  a'a  are 

cy,       —  ejS,       —  ea 

a         respectively.     Now 

AC  =  AC  +  CO  =  AB  +  BO, 
ft  -  Xey  =  y  +  ?/e/?. 


B     or 


Operating  witli  X  S  .  |S,  we  have,  since  ySeft-  =  0, 
HaB 

Having  assumed  o  to  be  the  intersection  of  the  altitudes  bb' 
and  ccj  let  o'  be  the  intersection  of  aa'  and  cc;     Then 

AO'  =  AC  +  CO^ 

or 

Zea  =  ft  —  x'ey. 


Operating  with  X  S  .  a 


Seya       Sacy 
_  ^fta         _ 


S(y-^)ey 
Haft 
S.yft' 


Haft 
-Hftey 


Hence  o  and  o'  coincide,  and 


GEOMETRIC    MULTIPI 

2.     To  rnrnmscribe  a  circle  about  a  trionijle 
T.ot  (Fig.  (Jl )  AC  =  ^,   cii  =  tt,   A 15  =  y. 
Then 

a'o  =  —  Xea, 
C'o  =  ?/€y. 

C)i)eratin;j,-  with  X  S  .  /J  on  the  expres- 


.ICATIOX    AND    DIVISION.         109 


u>  =  ^  y  +  Z/«y  =  i  /3  —  '-e/S' 


we  have 


y- 


Sa/3 


2S.y^ 
Operating  with  X  S  .  a  on 

ho'  =  —  ^  y  +  v'ey  =  —  .V  a  —  .X''ea, 

Sa^    ^  Sa/3 

2  Ssya  2  Ssy/3 


we  have 


Therefore  ?/ =  ^v'  and  o  and  o'  coincide. 
The  radius  nuiv  1h'  found  l)y  scjuariui 


Sa/ 


whence 


4       4&-c-siirA 
since,  if  a,  b,  c  arc  the  tensors  of  a,  (3,  y, 


ey, 


S(^y.2/^y)  =  o, 


,  a-Vrcos'-c 
4  />-c-  sin'-  A 


Hence 


yr  sin-  a  +  a-  cos-  c 


2  sin  A 


110 


QUATERNIONS. 


3.  In  any  triawjle,  the  centre  of  the  circumscribed  circle,  the 
intersection  of  the  altitndes  and  the  intersection  of  the  medials  lie 
in  the  same  straight  line;  and  the  distance  betiveen  the  last  tiro 
points  is  two-thirds  of  the  distance  betiveen  the  first  tv:o. 

Let  M  (Fig.  (J2)  ])L'  the  intersection  of 
Fig-,  ni.  |.jjj.  medials,  a'  that  of  tlie  altitudes,  and 

c  the  center  of  the  circle. 

Then,  from  Ex.  T),  Art.  11,  where  (P 
(Fig.  1 1 )  is  given  in  terms  of  the  adjacent 
sides,  we  have 

AM  =  i(/i  +  y). 

From  Ex.  1,  Art.  57, 


From  Ex.  2,  Art. 

But 

c:\i  = 

and 


-'  =  ^^^,- 


So/ 


S.y/3' 


AM=|/3-iy+Me^. 
Sey/? 

ma'=  2  cm, 


and,  since,  as  vectors,  they  are  multiples  of  each  other,  and  have 
a  common  point,  they  form  one  and  the  same  straight  line, 

4.  To  find  the  condition  that  the  iierpendiculars  from  the  angles 
of  a  tefraedron  to  the  opposite  faces  shall  intersect. 

With  the  notation  of  P^ig.  52,  the  perpendiculars  from  a  and  i? 
on  the  opposite  faces  are 

V/?y     and     Vya. 

If  the}'  intersect,  at  p  sa^',  then  must  a,  b,  p  lie  in  one  pUuie. 
Hence,  Art.  55,  4,  Cor.  3, 


[(/?-a)V/3yVya]=0, 


GEOMETRIC    MULTIPLICATION    AND    DIVISION.         Ill 


S  (iS  -  u)  [S  .  V/iy Vya  +  V  .  V/iyVya]  =  0 
S(/i-a)y.\>/Vyu  =  U. 

But,  P^qiuition  (117), 

V  .  \fiy\ya  =  -  yS/3ya  ; 
.-.  -(S/Jy-Sa7)S/3ya=0, 

I' 

S/3y  =  Say.  ((()        ( 

Froui  the  figure,  we  have 

BC-  +  OA-  =  {y  —  (3)-  +  a- 

=  r-->Sy(i  +  ft'  +  a' 


or,  from  (a), 


y-  -  -2  Say   +  ft- +  a' 
:  AC-  +  OB-. 


Hence  the  condition  is  that  the  sums  of  the  squans  of  each  pair 
of  oppositi'  edges  shall  be  the  same. 
o.    Interi)reL  Equation  (118), 

hSa(3y  =  \(3ySaS  +  VyaS/3S  +  Va/3Sy8, 

under  tlie  condition  that  a.  (3.  y  be  (•oui[)lauar  with  8. 

If  a.  /i,  y  are  complanar.  S://3/  =  0.  and  therefore,  8  being  in 
or  out  of  the  phine, 

Sa8  V^  y  +  S/3  oY  /a  +  S  yS  Va^  =  0 .  (  «  ) 

If  8  be   in  the  plane,   we  have  for  any  four  co-initial  lines 

OA,  OB,  OC,   CD, 

sin  Boc  cos  Aoi)  +  sin  coa  cos  bod  +  sin  aob  coscoi>  =  0, 
and,  for  a  line  i)erpendicnlar  to  on, 

sin  HOC  sin  aod  +  sin  coa  sin  bod  -f  sin  aob  sin  cod  =  0. 

If  8  is  perpendicular  to  the  plane,   the   ti'ruis   in  («)   vanish 
separately. 


112  QUATERNIONS. 

G.    If  X,  Y,  Z  he  the  angles  made  hy  any  line  op  vnth  three 
rectangular  axes,  then 

cos-  X  +  cos-  Y  +  cos-  Z  =  1. 
From  Equation  ((57) 

ip  =  xi-  +  yij  +  zik  ^  —  x  +  yk  —  zj, 
whence 

Operating  in  a,  similar  manner  witli  S  ../ X  and  S  .  A-  X  we  obtain 

If  Tp  =  ?•,  tlien  p-  =  —  r ,  Sip  =  —  r  cos  X,  etc.     Hence 

Op-  =  op-  (cos- X  +  cos-  Y  +  cos- Z) , 

or 

cos-  X  +  cos-  Y  +  cos-  Z—\. 

Applications  to  Spherical  Trig-onometry. 

Let  ABC  (Fig.  6o)  be  any  splierical  triangle  on  the  surface  of 

a  unit  sphere  whose  center  is  o  ;  a,  8.  y 

Fig.  63.  ,    .  .  „  .  .       '^ 

c'  benig  unit  vectors  from  o  to  the  vertices. 

I  The  sides  ab,  bc,  ca  represent  versors 

I                c  whose  angles  are  c,  a,  6,  and  axes  are 

L<::r!!rr_L\.      b'  <^>c'=y,'     OA'  =  aJ     (>i5'=/5';      a,    /3',    y' 

/v" ^1^    y*  being  unit  vectors  to  the  vertices  of  the 

a'/      \      /i^'y^  polar  triangle  Avhose  sides  are  a',  h\  q\ 

/             \A. — ^  the  supplements  of  the  opposite  angles 

a'  a,  b,  c  of  the  triangle  abc. 

7.    We  have  first  ^      ^ 

7     "  y 


GEOlSrETEIC    ^rULTTrLTCATION    AM)    DIVISION.        113 
Taking  the  sralnrs.  we  have  [Kcniatioii  (!)0)], 


But 


and 


(b) 


B  B      a  R      a 

S'-  =  S-  S-  +  S  .  V-  V-. 

y  a      7  (r      y 

13  (3  a 

y  a  '  y  ' 

=  sine  sin^  ^y'(3' 

=  —  sine  sin  6  cos  a' 

=  sine  sin  6  cos  a. 
Hence,  in  («), 

cos  (I  =  cos  c  cos  /^  +  sin  c  sin  ^j  cos  a. 
B}'  a  cyclic  permutation  of  the  letters  in  («),  we  obtain 

a       13    a 
Whence,  as  before 

a  fj      a  13      a 

or 

eos&  =  cos  a  cose  +  sinr^  sine  Sa'y', 

in  which  Sa'y'  =  —  cos?>'  =  cos  p.. 

.  ■ .  cos  b  =  cos  u  cos  c  +  sin  a  sin  c  cos  b.  (c) 

Similarly,  or  directly  by  cyclic  permutation  in  (c), 

cose  =  COS&  cos  a  +  sin  b  sino  cose. 
From  the  relation 

f3'  ^ft'   a\ 

7     <^'  y' 

ma}'  be  deduci'd  in  like  manner 

—  cos  A  =  cose  cosii  —  sine  sinn  cosce. 


11-t  QUATERNIONS. 

8.    Resuming  the  equation 

y        ay 

of  the  last  example,  and  taking  the  vectors,  we  have  [Equa- 
tion (!»1)], 

v^  =  s^-  +  s-y^  +  Y.v^y-.  (-) 

yayytt^  ay 


But 


7 
V  .  Y 


V-  =  —  a  snia, 

7 

13     a 

S- V-  =  cosc(^'sin6)=:coscsin?>  .  (3', 

a     (i 

S-V- =  eos6  (y' sin  c)  =  COS  ?>  sine  .  y', 

-  V~  =  V(y'sin(')  (y8' sin ?>)=  sine  sin6  Vy'/3' 

=  sin  e  sin  h  ( —  a  sin  a')  =  —  sin  e  sin  h  sin  a  . 


Substituting  in  (a), 

—  sin a  .a'=coscsin6,/?'  +  cos6sinc»y'— sincsinisinA.a.  {h) 
Operating  with  x  S  .  y'~\ 

—  siuff  .  S-,  =  coscsiniS— +cos6sinoS^,-sincsin6sinAS-,. 

7  7  7  7 

in  which 

S  — ,  =  cos  b'  =  —  cos  B, 
7 

»  — ,  =  —  cos  A , 

7 
S  -,  =  0,  since  a  and  y'  are  at  right  angles. 

Hence 

sin  a  cos  b  =  cos  b  sine  —  cose  sin  b  cos  a, 


GEOMETRIC    MULTIPLICATION    AND    DIVISION.        115 
aiul  ill  the  same  manner,  or  by  a  cyelie  permutation  of  the  letters, 

sin  h  cos  c  =  cos  c  sin  a  —  cos  a  sin  c  cos  b, 
sin  c  cos  A  =  cos  o  sin  h  —  cos  h  sin  a  cos  c. 

9.    Operating   on    Equation    {h)    of    the    hist   example   with 
X  V  .  y'-'  instead  of  X  S  .  y'"', 

—  sin  a  V  -,  =  cos  c  sin  ^  V  -,  +  cos  h  sin  c  V  -,  —  sin  c  sin  h  sin  a  V  -,• 

y  y  y  y 

But 

(  Fig.  03. 


V -^  =  /3  sin  ?>'  =  /?  sin  B, 

y 

c' 
1 

V^,  =  —  asin«'  =  —  asiuA, 

y 

1 

y 

/ 

>,;,=... 

7 

Substituting  these  ^-alues 

—  sin  a  sin  B  .  /?  =  —  cos  c  sin  h  sin  a  .  a  —  sin  c  sin  h  sin  a  .  /s! 
Dividing  by  a,  and  substituting  for 


cos  c  +  y'  sin  c     and  for  —  =  yj 


we  obtain 


—  sin  a  sin  b  cos  c  —  sin  a  sin  b  sin  e  .  y '  =  —  cos  c  sin  h  sin  a 
—  sin  c  sin  h  sin  a  .  y'. 

Equating  the  scalar  or  vector  parts,  we  have  in  either  case 

sin  a  sin  b  =  sin  a  sin  &, 
or 

sinrt  :  sin?> :  :  sin  a  :  siiiB. 

The  formulae  of  the  preceding  examples  have  all  been  deduced 
from  tlie  equation  -  =  -  —      The  i)roduct  as  well  as  the  quotient 

y        a  y 

may  also  be  employed,  as  follows : 


116  QUATERNIONS. 

10.  Assuming  the  vector  product 

and  taking  tlie  vector  part,  we  have  [Equation  (117)], 

y.Ya/3\(3y=^-/3Ha(3y.  (a) 

But 

V  .  Vay8V/?y=  V(y'sinc)(a'sina)  =  sine  sina  sinii  .  /3, 

and.  Art.  55,  4, 

Sa;8y  =  —  sine  sin^J 

6'  being  the  angle  made  by  oc  with  the  plane  of  c.     Substituting 
in  (a), 

sine  siutt  sinii  .  fS  =  sine  sin^' .  /S, 

Fig.  03. 

or 

sin^' =  sinrt  siuB. 

By  permutation,  from  (a), 

,     X  V  .  \ya\a/3  =  —  aSya/S  =  —  aSa/3y, 

'  iT^  yii      or 

sin^  sine  sin  a  .  a  =  sine  sin^' .  a, 
a'  .-.  sin^'  =  sin/>  sin  a. 

Equating  these  values  of  sin  ^'  we  have,  as  in  Example  9, 
sin  a  :  fi'mb  :  :  sin  A  :  siuB. 

11.  Let  p„,  j)f,^  2')^  represent  the  arcs  drawn  from  the  vertices 
of  ABC  perpendicular  to  the  opposite  sides. 

Resuming  Equation  (a)  of  the  preceding  example,  and  taking 
the  tensors, 

TV  c  Va^V/3y  =  Sa(3y  =  sine  sin^),, 
=  S/3ya  =  sin  a  sin^7„, 
=  SyafS  —  sin  b  sin2>i, 


GEOMETIUC    ]MULTIPLICATION    AND    DIVISION.        117 

and,  takin<i-  i\\c  U-nsnv  of  Y  .  \uf3\fty  IVoin  tlic  last  example, 

siuo  sin  a  sin  n  =  sin  a  sin />„  =  suib  sinji>j  =  sine  sin/><., 
or 

sinj>„  =  sine  sinn, 
sin  r  sin  a    . 


smjh 


sin  6 

sin^;^  =  sin  a  sinB. 

12.  Show  that  if  ahc,  a'ij'c'  be  ta-o  tri-rectaiigidar  triaiujles  (t)L 
the  surface  of  a  s})here, 

cos  aa'  =  cos  uu'  coscc'  —  cos  b'c  cos  bc', 

the  triangles  being  lettered  in  the  same  order. 

Let  a,  y8,  y,  a',  (S',  y'  be  tlie  vectors  to  the  vertices.  These 
being  at  right  angles,  in  eacli  triangle,  we  have 

COSAa'  =  —  Stttt'  =  —  S  .  y/3yV/3'y! 

or,  Equation  (122), 

cos  aa'  =  S^yS'Syy'  -  S/3'yS/3y' 

=  cos bb'  cos €'(,'  —  cos h'c  cos bc'. 

[The  vectors  of  P2qnation  (122)  are  arbitrary,  but  we  ma}' 
divide  both  members  by  the  tensor  of  the  product  of  the  vectors, 
so  that 

S(VUa/3VUy8)  =SUaSSlT/5y  -  SUaySU/38, 
for  the  unit  sphere.] 

13,  Let  ABCi)  be  a  splierical  (juadrilateral  whose  sides  are 
AB  =  a,  BC  =  b,  m  =  e,  da  =  d,  tlie  vectors  to  the  poles  of  these 
arcs  being  a,  ft',  yj  8'  respcctivel}-.     Tlien 

Va/8  =  a' sin  a, 
VyS  =  y'sinc. 


118  QUATERNIONS. 

From  Equation  (122), 

S  .  VaySVyS  =  Sa8S/?y  -  SayS/58, 

or  . 
sma  sine  Sa'y' =  (—  cos  da)  (  —  cosbc)  —  (— cosdb)  (  —  cosac). 

But  Sa'y' =  — cos  L,  L  being  the  angle  formed  by  the  ares  au 
and  CD  where  they  meet,  the  ares  being  estimated  in  th  > 
directions  indicated  b}'  the  order  of  their  terminal  letters. 
Hence 

siuAB  sin  CD  COSL  =  COSAC  COSUD  —  cos  AD  COSBC, 

a  formula  due  to  Gauss. 

14.  Retaining  the  above  notation,  abcd  being  still  a  spherical 
quadrilateral,  denote  the  angles  at  the  intersections  of  the  arcs 
AB  and  CD,  AC  and  db,  ad  and  bc,  bj-  l,  m  and  N  respectively. 
Then,  from  Equation  (125), 

S[Va;8Vy8  H-  VayVSyS  +  VaSV^y]  =  0, 

we  have  identically 
sin  AB  sin  CD  eosL  +  sin  ac  sIubd  cosm  +  sin  ad  sin  bc  cosx  =  0. 

Were  the  points  a,  b,  c,  d  on  the  same  great  circle,  the  angles 
L.  :m  and  n  would  be  zero,  and  the  above  reduces  to 

sin  AB  sin  cd  +  sin  ac  sin  bd  +  sin  ad  sin  bc  =  0. 

and  for  a  line  oaJ  perpendicular  to  oa  and  in  the  same  plane, 
dropping  the  accent,  we  have 

cos  AB  sin  CD  +  cos  AC  sin  bd  +  cos  ad  sin  bc  =  0, 

which  are  the  results  of  Example  5  of  this  article. 


GEOMETRIC    MULTIPLICATION    AND   DIVISION.        119 

58.   General  Formulae. 

1.  We  have  seen,  Equation  (SG),  that  S2  =  2S  and  V2  =  2V  : 
but  (Art.  50,  4)  that  2T  is  not  equal  to  TS,  nor  2U  to  U2.  We 
have  also  seen.  Equations  (9G)  and  (*J7),  that  Til  =  IIT  and 
Un  =  nU;  but  SII  is  not  equal  to  IIS,  nor  Vn  to  nV :  for. 
1st,  SII  is  independent  of  the  factors  under  tlie  n  sign,  provided 
the  product  remains  the  same,  wliile  nS  is  dependent  upon 
them  ;  and.  2d,  because  (Art.  55,  5)  IIV  is  not  necessarily  a 
sector. 

2.  Resuming  Equation  (92), 

S/vy  =  Sr/r, 

and,  since  r  is  arbitrary.  Avriting  rs  for  ?*,  we  have.  l\v  the  asso- 
ciative law  (Art.  52), 

S(As)f/  =  S7(rs), 

Sr(6'g)=S(.sry)r, 
.  • .  Hrsq  =  S.sgr  =  Hqrs  .     .      .     .     ( 1 2G ) , 

a  formula  which  may  evidently  be  extended.  Hence,  the  scalar 
of  the  product  of  any  number  of  quaternions  is  the  same,  so  lo)iy 
as  the  cyclical  ordi'r  is  maintained. 

3.  Let  yi,  q.  r.  s  l)e  four  quaternions,  such  that 

qr=ps.  (a) 

Operating  witli  K7  X  , 

K7  .  (//•  =  (K7  .  7)  r  =  (qKq)  r  =  K7  .  ps. 

since  conjugate  quaternions  are  commutative.     Hence 

{Tqy-r  =  Kq.ps, 
or    . 

Kq  .  ps  1  /I.)-, 

(T7)-  ^    ^         7    ^ 

Operating  on  {a)  with  xKr.  we  have 

qr  .  Kr  =  2)S  .  Kr, 


120  QUATEKNIONS. 


q(Try  =  j^sKr, 
psKr  1 

Hence,  in  any  equation  of  the  x>'>'oducts  of  tivo  quaternions, 
the  first  factor  of  one  member  may  he  removed  by  writing  its  con- 
jugate as  the  first  factor  of  the  second  member,  and  dividing  the 
latter  by  the  square  of  the  tensor,  or  simply  by  introducing  the 
reciprocal  as  the  first  factor  in  the  second  member.  By  substi- 
tuting the  word  last  for  first,  the  above  rule  will  api.>lv  to  the 
second  transformation. 


4.    Resuming,  for  facilit}'  of  reference,  the  equations 

q=~  =  ^{GOScl>-\-esmct.)  =  Tq.Vq  =  Hq  +  yq, 

(^1) 

1       1       ^      T;8 
g-i  =  -  =  -  =  ^(cos</>-esmc/>), 

(^) 

Kg  =  ^  (cos ./.  -  6  sin  </>)  =  S(/ -  Vg, 

(C) 

we  observe  directly  that 

Sq  =  S  (Try  .  Vq)  =  Try  .  SUr/     .      .      .     (1 2a) , 

Yq^T\q  .VYq  =  Tq  .YVq      .      .      .      (130), 

TYq  =  Tq  .  TYVq  =  TVKry     .      .      .      .     (131 ) . 

5.  It  has  been  already  shown  (Art.  54,  Fig.  40)  that 
(Ta)2-|-  {Tf^)-=  (Ty)-,  and  (Art.  54,  Fig.  42)  that  Ta  =  Ty  .  cos</), 
T^  ==  Ty  .  sin  ^  ;  and  therefore 

(Ty)2cos^c^  +  (Ty)2siir<^  =  (Ty)S 
or 

sin-0  +  cos"(/)  =  I. 

Hence,  from  Equations  (44), 

(SUg)-  +  (TyiTry)2  =  i     ....     (132). 


GEOMETRIC    MULTIPLICATION    AND    DIVISION.        121 

This  iini)()rt:iiit  fonmihi  might  have  been  writtt'ii  ;it  oiiee  l)y 
assuiniiig  the  til)ove  well-known  relation  of  Phuie  Trigonometry. 

G.    From  Equations   (129)   and   (131),  we  may  write  Equa- 
tion (132)  under  the  form 

(S./)'-'  +  (TV<y)-'  =  (T5)2 (133), 

or,  from  Equation  (107), 

{iiqy-{\qy  =  {Tqy  =  (Sqy-+{T\qy       .     (134), 


7.  Comparing  {A),  (B)  and  (C), 

SUg  =  SU-  =  SUKg    .     .     .     (135), 

TVUg  =  TVU-  =  TVUK7      .     .     (136), 
and  from  Equations  (129)  and  (135), 

Sq  =  Tq  .  SUry  =  Tg  .  SU-  =  Try  .  SUKg  .     .     (137). 

8.  Since  Tq  =  TKg,  we  have 

Tq.TKq  =  {Tqy (138), 

and  Try  being  a  positive  scalar, 

KTq  =  TKq (139). 

As  exercises  in  the  transformation  of  these  and  the  following 
sjTubolical  equations,  some  of  the  results  already  obtained  will 
be  deduced  anew.  Thus,  to  prove  that  T{qq')  =  TqTq\  whence 
T  .  q-  =  {Tqy,  we  have 

T(77')'  =  (w')K(ryg')  Equation  (107) 

=  qq'Kq'Kq  Equation    (99) 

^q(q'Kq')Kq  =  {Tq'yqKq 

=  iTq'y(Tqy, 

•••  T(qq')  =  TqTq: 


122  QUATEKNIONS. 

9.  Substituting  for  Sq  and  T\q  their  values  from  P^quations 
(79)  and  (131) 

{SK<jr+iT\Kqy-  =  {^qy-  +  {T\qy     .      .     (140). 

10.  Resuming  from  Art.  51,  1,  the  expressions 

■  \rq  =  SrVg  +  SqYr  +  V  .  V/Tg,  (a) 

\qr  =  SfyVr  +  SrVfy  +  V  •  VgVr,  (b) 

Sqr  =  S(/Sr  +  S  .  VgVr,  (c) 

we  liave,  by  adding  and  subtracting, 

Yqr  +  \rq  =  2  Sry  Vr  +  2lSrTg  1  ,^^^. 

yqr-Nrq^2\  .yqyy  |       •      •      •      •      ^        ;• 

And,  if  g  =  r,  from  (a)  and  (c), 

V.(/-  =  2SryVg  I  .j^2) 

S.r/  =  (Sry)2  +  (Vry)2| 
whence 

q'  =  {^qf  +  2^qJq  +  {\qY.      .      .      (143). 


SU.r/^(SUg)^  +  (VUg)^  ^  ^j4^^ 


Dividing  Equations  (142)  by  (Try) 

SU.r/==(SUfy)2  +  (^ 
VU  .q-  =  2  SUg  .  VUg 
since,  evidently, 

S.r  =  (Tg)^SU.r)  .145. 

V.g2  =  (Tg)"VU.g-i 

Again,  substituting  in  the  second  of  Equations  (142)  the  value 
of  (Yg)-  from  Equation  (134),  we  have 

S.g^=2(Sg)— (Tg)2     .....     (146), 

and  dividing  by  (Tg)- 

SU.g-  =  2(SUg)2-l (147). 

Substituting  (Sg)-  from  the  same  equation 

S.g^-2(Vg)^  +  (Tg)2 (148). 


GEOMETRIC    MULTIPLICATION    AND    DIVISION.        12:5 

K((iiations  (1  10)  and  (148)  may  be  wiittcn 

( s  4-  T)q-  =  2 ( Sv) -      and      (S  -  T) r/  =  2{\q) -. 

Iiitrodticing  in  (a),  or  (6),  the  condition  tliat  7   and  r  are 
couiplaiuir,  we  have,  after  substituting  versors, 

VUryr  =  VUfySU/'  +  VUrSUg, 

since,  under  the  condition,  V(VU9VUr)  =  0. 

Taking  the  tensors,  since  q  and  r  are  complanar, 

TVU^r  =  TVUgSUr  +  SU^TVUr  .     .     .     (149), 
and,  interpreting,  Art.  51,  6, 

sin(^  +  <^)  =  sin^  cos^  +  eos^  sin<^. 
Introducing  tlie  same  condition  of  eomi)lanarity  in  (c) 
Sryr  =  HqSr  -  T\qT\i; 
or,  substituting  versors  as  abOV^, 

Sir^r  =  SUgSUr  -  TVUfyTVUr    .     .     .     (150), 
or,  interpreting, 

cos (^  +  <^)  =  cos 6  cos  </)  —  sin  ({>  sin 0. 

11.    Putting  Equation  (140)  under  the  fbrm 


|S.r/  +  T.r/^ 


Sry 

and  writing  Vq  for  </,  we  have 

SV^  =  V^(S7  +  T7)       ....     (151). 
12.   Taking  the  tensors  of  the  first  of  Equations  (142).  we  have 

TV  .  q- 

■'  2Sq 


124  QUATERNIONS. 

and  writing  Vg  for  q 

TVo 

or,  by  Equations  (133)  and  (151), 


whence 
and 


TV     r    1   ((M!:^!^ 


S^) 
TV  .  Vr7=  V^(Tg-Sr/) (152), 


iTg-Sg 


(153), 


13.   F'rom  the  definition  of  the  powers  of  a  quaternion,  we  have 

g-'-fy-^l,     (g«y'  =  r     ....     (lot). 

Hence,  since  g  =  T(/  .  Ug,  TH  =  HT  and  Un  =  nu, 

Tq-^  .  Try™  =  1 ,      Ug-*"  .  IVy"  =  1       .      .     (1 55 ) . 

Also,  because  TJ^"'"  =  UKg'", 

g-'»  =  Tg-™  .  Ug-'"=  Tg-'"  .  UKg™  =  Tg-2"'Kg'% 

or,  since  'Kpq  =  KgKys,  writing  pg  for  q,  and  making  m  =  1, 

(yxy)-^  =  T{2yq)-"-K2oq  =  T{pq)-niqKp 
^ll{pq)-\Tq)\Tpfq-'p-\ 
Or 

{pq)-'  =  q-'p-' (15G), 

the  reciprocal  of  the  product  of  tivo  quaternions  being  equal  to  the 
product  of  their  reciprocals  in  inverted  order. 

This  fornuda  may  be  extended  by  the  Associative  principle,  by 
a  process  similar  to  that  employed  in  the  deduction  of  I^lquation 
(126),  so  that  if  n'  represent  the  product  of  the  same  factors  as 
those  of  n,  in  reverse  order, 

{Uq)-'  =  U'q-' (157). 


GEOMETRIC    MULTIPLICATION    AND    DIVISION.        125 

The  (Miuntioii  Kp</  =  Kq}i.p  ma}'  be  deduced  without  reference 
to  spherifiil  arcs.     For,  b}'  Art.  44,  any  two  quaternions  can  be 

reduced  to  the  forms  (/  =  -,  i>  =  ^,  whence 
a  /3 

and  therefore 

K/>  .  y  =  K/)  .  p/3  =  (K^)  .  j))^  =  {TpYfi. 
Now 

(KryKi))y=  Kg(Tp)^'/?  =  (T^>)^K7  .  (i 
^  (T^>)'-'K(/  .  qa  =  {TpYiTqya  =  {Tpq)' a 
=  lipq  .  j)(/  .  a    =  Kpq  .  y 
•■•   Kpq  =  KqKp, 

whicli,  by  the  Associative  law,  gives 

Kn  =  n'K (1;38). 

14.  Show  that  K(-fy)  =  -Kv. 

15.  Show  that 

T{p  +  qy  =  (p  +  q)(Kp  +  Kq) 

=  {Tpy  +  {Tqr  +  2S.pKq 

=  {Tpy  +  (T7)-  +  2TpTrySU  .  ^^K^ 

=  {Tp  +  Tqy  -  2Tj>Tq(l-SV  .  pKq) . 

and  therefore  that  'T'{p  +  q)  cannot  be  greater  than  the  sum  or 
less  than  the  difference  of  Tp  and  Tq. 

16.  Show  that  r/UVf/-'  =  TVf/  -  HqV\q. 

59.   Applications  to  Plane  Trigonometry. 
1.    For  formulae  involving  20.  \vt 

q  =  Try  (cos  20 +  €  sin  2  0) . 
Then 

\/q  =  q'^  VTry(cos(9  +  esin^). 


126  QUATERNIONS. 

From  Equation  (142) ,  S  .  (/-  =  (S7)'-  +  (Vry)%  '^e  then  have 

or,  dividing  out  Try, 

SUry-(SlV)^^-f(VUr/)^ 
and,  interpreting, 

cos  2  0  =  vos-0  —  shrO. 

Again,  from  Equation  (147).  SU  .  </- =  2(SU</)- -  1, 

S%  =  2(SUr/)--l; 
whence 

cos2^  =  2cos-^-l. 

Again,  from  Equation  (142),  V  .  q"=2SqYq, 

Vfy  =  2Sf/\V, 

or,  dividing  out  Tq  and  e, 

TYUfy  =  2  SUf/'TVUg' ; 
whence 

sin  2^  =  2  cos  6  sin^. 

2.   Resuming  Equations  (149)  and  (150), 

TWqr  =  TVUfySUr  +  SUgTVUr, 
SU(/r  =  SUrySUr  -  TVU^TVUr, 

which  have  alread}-  been  interpreted  as  the  sine  and  cosine  of 
the  sum  of  two  angles,  and  writing  for 

'/•  =  Tr(cosc^  +  €sin<^),     r"^  =  — (cos<^  —  csin^), 

q  and  r  being  complanar,  Ave  have 

TVUryr-^  =  TVUgSUr  -  SUryTYUr       .      .     (159), 
SUryr-i  =  SUrySUr  +  TVlVyTVUr       .     .     (160), 
or,  interpreting, 

sin  (0  —  cfi)  =  sin  0  cos  <;^  —  sin  <^  cos^, 
cos  {6  —  (f))  =  cos 0  cos  (j>  +  sin 0  sin  </>. 


GEOiMIOTIIK;    MULTIl'I.ICATION    AND    DIVISION.        127 

3.  Acldiiio-   iMiuatioiis  (11!))  iiiul  (151)), 

'r\{](jr  +  TVUv/--'  =  2  SUrTVUry, 
in  which,  if  qr  =  p,  (jr-^  =  t,   .-.  q=\/pt,  r  =  ^/ptF^, 

TVUy)  +  TVU/  =  2SU(Vi^')TVU(Vi^)      .     (ir,i), 
or 

sin  X  +  sin  ?/  =  2  cos  ^  (x  —  ?/)  sin  i  (x-  +  y) . 

Similarly,  by  subtracting  the  same  ccjuations, 

TVU^r  -  TVUr//--'  =  2  SUvTVUr, 

TVU/)  -  T VU;  =  2  SU ( Vpt) TVU ( Vi^' )  .     ( 1 G2) , 
or 

sin  .X"  —  sin  ?/  =  2  cos  4^  (.«  +  ^'/ )  sin  ^  ( .i'  —  // ) . 

4.  I'roni  Equations   (150)   and   (ir)()),by  addition  and  sub- 
traction, we  oljtain,  in  a  similar  manner, 

SVj>  +  SVt  =  '>SV{Vpt)^V{Vpt-')     .     .     .     (1G3), 
and 

SUyj  -HVt  =  -2  TVU(  ViTOTVU(  VvTr ') , 
whence 

cos  a-  +  cos?/  =  2cos^(.f  +  ^)  cos^(x-  —  .'/), 
cos  1/  —  cos  X  =  2  sin  ^{x  -\-  y)  sin  ^  (•«  —  ?/)  • 

5.  Resuming  Equation  (152), 


TVV7  =  VKTv-!^7), 
it  may  be  put  under  the  (brni 

2(TyuV7)'  =  l-SUry, 
or 

2sinH^^l-cos^. 

and,  in  a  shuilar  maimer,  from  Ecjuation  (151), 

Sy^  =  Vi(Sry-fTry), 
2(SUV(/)-  =  SLVy  +  l, 
or 

2eosH^  =  l  +  cos^. 


1'28  QUATERNIONS. 

6.  From  Equation  (142) 


(TV:S)r  =  ^MI^ 

{»(/)'+ {yqy 

_2TVq  (Sf/)- 

Sq     '  (Sg)^-(TVg)2 
2 (TV  :  S)g 


tau  2  ( 


l-[(TV:S)cyr 
2  tan  ^ 


1-tan-^ 
And,  in  a  similar  manner, 

cot-^-1 


cot  2  ^  : 


2cot^ 


7.  From  Equations  (90)  and  (91),  q  and  r  being  complanar, 

Sgr  =  SrySr  +  S  .  Vy Vr  -=  SgS/-  -  TV^yTVr, 
TVfyr  ^  Sr/TVr  +  SrTVg, 

we  ha\e,  by  division, 

(TY;S)r,.^^-/'ry>-+S.Ty, 
SfySy  -  TVryTVr 
^  (TY;S)r  +  (Ty;S)ry 
1-(TV:  S)y(TV:  S)r 


Also 


4.      /  /I  ,    J  \       tan  4>  +  tan  ^ 
tan(^  +  <^)  =- — 7^ 

1  —  tan  <i  tan  6 


^  1+(TV:  S)ry(TV:  S)r 

,       ,  ^        , .        tan  ^  —  tan  d> 
tan(^  -</>)  =  -— ^ — TV"^- 
1  +  tan  6  tan  (^ 


8.  From  Equation  (153) 


Try  -  Sq 


^  \Tq  +  Sq 


GEOMETRIC    MULTIPLICATION    AND    DIVISION.        129 

by  substitiitiiiij;   in   succH'ssion   p   and   I   for   V(/,    .•.   'J  =  p'  :iii(l 
q  =  t-.  we  obtain,  alliM-  i-t'cUu'tion, 

TV/>S/  +  T\mp 


(TV  :  S)^;+(TV:  S)t  = 


SpSt 


or,  dividing  out  Tp'Vf   from  the  second  member,  and  applying 
Equation  (149), 

{T\:H)p+{T\:H)t  =  ^P^ 

or 

sin(a;  +  ?/) 

tan  a;  +  tan  y  =  — ^ ^-. 

cos x"  cos?/ 

And  similar!}-,  by  subtraction  and  api)l3ing  P^quation  (159), 
(TV:  H)p-  (TV:  S)t-'^^^^^^' 


SVpSVt 
or 

sin(a;  — ?/) 

tan  x  —  tan  y  = ^^ ^^. 

cos  a*  cos?/ 

9.  From  Equations  (IGl)  and  (163) 

TWV pt  = ^^, 

2  su  Vi>ri 

'''^'''"''^  -  -      TVr»4-TVlV 

(TyU:SU)V,.  =  (TV:S)V,.  =  ^^^^^±|li^, 

or  .       ,    . 

,      1  /     ,     \       snia-  +  sm?/ 

tan  ;V(.r  +  y)  = -^ . 

cos  or  +  cosy 

And,  in  a  similar  manner,  from  Equations  (102)  and  (1().S), 

(TV:S)V,'^  =  'r^'^^^--^^'^'^ 
^  '     ^  HVp  +  ^Vt 

or 

tan^a--?/)       sin  a.- sin  ?/ 


cos  x  + cos// 


130   •  QUATERNIONS. 

10.  Similar  formulae  ma}-  be  deduced  for  functions  of  other 
ratios  of  an  angle.  Thus,  from  Equation  (90),  writing  rs  for 
/•,  and  making  q  —  r  =  s  all  complanar,  we  have,  by  Equation 
(142), 

ii.<r  =  {Sqr-3Hq{TYq)\ 
or 

cos  '3  6  =  t'os^O  —  3  cos^  sin-^, 

or,  under  the  more  familiar  form, 

cos3^  =  4cos^^-3cos^. 


CHAPTER   III. 
Applications   to    Loci. 

60.  Any  vector,  as  p,  ma}'  be  resolved  into  three  component 
vectors  i)arallel  to  any  three  given  vectors,  as  a,  /8,  y,  no  two 
of  which  are.  parallel,   and  which  are   not   parallel  to  any  one 

plane.     Thus                                ,     «  ,  nri\ 

^  p^xu  +  ;/(^+zy (164) 

refers  to  any  point  in  space. 

If  the  variable  scalars  x,  y,  z  are  functions  of  two  independ- 
ent variable  scalars,  as  t  and  n,  p  is  the  vector  to  a  surface, 
which,  if  the  functions  are  linear,  will  be  a  plane.     We  may, 

therefore,  write                             ,.,      x  /ir-\ 

p  =  <^(^  u) (16a) 

as  the  general  equation  of  a  surface. 

If  X.  y  and  z  are  functions  of  one  independent  variable  scalar, 
as  ^  p  is  the  vector  to  a  curve,  which,  if  the  functions  are 
linear,  becomes  a  right  line.     We  may,  therefore,  write 

P  =  4>{t) (ICO) 

as  the  general  equation  of  a  curve  in  space. 

If  a,  /8,  y  are  complanar,  we  may  replace  either  two  of  the 
vectors  in  Equation  (164)  by  a  single  vector,  in  which  case 
p  =  <f){t,)  contains  but  two  variable  scalars,  functions  of  f.  and 
is  the  equation  of  a  plane  curve,  or  of  a  straight  line  if  the  fum-- 
tions  are  linear. 

The  essential  characteristic  of  the  various  equations  of  a 
straiglit  line  is  that  they  are  linear,  and  involve,  exi)licitly  or 
implicitly,  one  indeterminate  scalar. 


132  QUATERNIONS. 

61.  Assuming 

p  =  xa  +  yfB,  (rt) 

in  wliifli  X  and  y  are  variable  scalars,  functions  of  a  single  A'ari- 
able  and  independent  scalar,  as  i,  as  the  general  form  of  the 
equation  of  a  plane  curve,  by  substituting  in  any  particular  case 
the  known  functions  x—f(t),y=f'{t),  or  x=f"(y),  we  may 
avail  ourselves  of  the  Cartesian  forms  and  apply  to  the  resulting 
function  in  p  the  reasoning  of  the  Quaternion  method. 

For  example,  suppose  a  and  /3  are  unit  vectors  along  the  axis 
and  directrix  of  a  parabola,  the  origin  being  taken  at  the  focus. 
In  this  case  we  have  the  Cartesian  relation 

y-  =  22)x  +  jr,  (&) 

or,  substituting  in  (a), 

P  =  — ^  {ir  -  V-)  a  +  yf3, 

as  the  vector  equation  of  the  parabola. 

Or,  again,  a  and  /?  being  any  given  vectors  parallel  to  a  diam- 
eter and  tangent  at  its  vertex, 

P  =  fa  +  /^  (c) 

is  the  vector  equation  of  a  parabola,  in  terms  of  a  single  inde- 
pendent scalar  t. 

62.  Let/(.i-)  be  any  scalar  function  as,  for  example, 

Then 

dlf(x)]=2xdx  =  [f'(x)](lx. 

If,  however,  /(q)  be  a  function  of  a  quaternion  q,  as,  for 
example,  in  the  above  case, 

then 

f(/J  +  dq)  =  {q  +  dqY-  =  q-  +  qdq  +  dq  .  7  +  {dq)\ 
■■  ■  (lLt\^l)li  =  Qdq  +  dq  .  q. 


Al'l'LICATIONS    TO    LOCI.  133 

which  cfinnot,  Iiowcvcr.  Ix'  written  'Iqdq,  bocmise  of  the  non- 
comimitative  chanuU'r  of  quaternion  niultii)lic'ation.  We  can- 
not, therefore,  write,  in  general, 

<?[/(7)]  =  [/'(7)]^?'?, 

or  form,  as  usual,  a  differential  coefficient.  Since  vector,  as 
well  as  quaternion,  nuiltiplication  is  non-commntative,  the  same 
is  true  of  the  dilferentiation  of  a  function  of  a  vector.     Thus,  if 

f{p)  =  p\ 

and  in  order  to  wa-ite  f?[/(p)]  =  \_f'{ii)'\(1p.  it  would  be  necessary 
to  determine  a  vector  «-,  such  that  ud;)  =  dft  .  p,  or 

o-  =  dp  .  pdp'^, 

or,  if  £  be  the  vcrsor  of  dp,  since  the  tensors  cancel. 


that  is  (Art.  56,  18),  we  must  have  p,  e  and  o-  complanar,  or 
Yea  =  V/oe.  Since  complanar  quaternions  are  commutative,  if  q 
and  dq  are  complanar,  or  if  dq  or  dp  is  a  scalar,  this  peculiarity 
of  quaternion  and  vector  differentiation  disappears.  In  this 
case,  dq  and  dp  being  scalars,  f{q)  or  /(p)  are  quaternion  or 
vector  functions  of  scalar  variables,  to  which  the  ordinary  rules 
of  differentiation  are  applicable.  In  fact  we  have  only  to  assume 
such  a  function,  as 

p  =  x'a'  +X-"a"  -\-x"'a"'  + =  S.ra  =  (f>(t)  , 

in  which  a',  a",  a'",  arc 'constants  and  the  only  \ariables  are 

the  scalar  multipliers,  to  sec  that  the  vectors  a',  a",  a'" are 

to  be  treated  as  constants  and  the  usual  rules  of  differentiation 
applied  to  the  scalar  coefficients. 

Such  equations,  then,  as  those  of  the  paral)ola,  {!>)  and  (f), 


134  QUATERNIONS. 

Art.  Gl,  in  which  a  and  ^8  are  given  constant  vectors,  ma}'  be 
differentiated  as  usual.     TIius,  from 

we  have 

p  and  p  being  anv  two  vectors  to  the  curve, 

p'— p  =  Ap 

is  the  vector  secant ;  so  that  when  p  and  p  become  consecutive, 
and  the  secant  a  tangent, 

dp  =  {ta+(i)dt 

is  a  vector  along  the  tangent  to  the  curve  at  the  point  corre- 
sponding to  t.     The  vector  to  this  point  being  ^^a  +  t^,  and  x 

any  variable  scalar,  we  may  write  the  equation  of  the  tangent 
line  at  that  point 

p  =  ^-a  +  t/3  +  x(ta-\-(3); 
for  any  given  point,  x  he'ing  the  only  scalar  varial)le. 

63.  It  has  been  seen  that  the  usual  definition  of  differential 
coefficients  is  inapplicable  to  quaternions  in  general,  for  this 
definition  involves  the  commutative  i)roperty  of  multiplication, 
which  is  not,  in  general,  true  of  quaternions,  nor  of  the  vectors 
to  which  they  may  degrade.  It  becomes  necessar}-,  therefore,  to 
give  a  definition  of  differentials  which  shall  not  involve  this  prop- 
erty, 3'et  which  shall  also  be  true  of  quaternions  which  degrade 
to  scalai's,  and  therefore  be  equally  applicable  to  ordinary  scalar 
quantities. 

If  iJ=/(7),  such  a  definition  is  involved  in  the  formula 

^U^  =  ,f'^'ln[f{q  +  n-hlq)-/(q)]     .      .      (167), 


APPLICATIONS    TO    I.OCI.  135 

for,  lct/(7,  r,  s,  )=^  'x^  nny  relation  between  a  system  of 

quaternions  q,  i\  .s,  ,  and  let  A7,  A?-,  A.s,  l)e  finite  and 

simultaneous  difTerences,  so  that  7+A7,  r  +  Ar,  .s  +  A.s,  

satisf\'the  relation ./'(v,  i\  .s,  )  =  (».    Thi-n  in  ijassing  from  tiie 

new  sj'stem  q  +  Aq,  to  the  old  system  7, (he  simul- 
taneous differences  can  all  he  m:id(>  to  approach  zero  together, 
since  they  all  vanish  together.     If,  while  these  differences  A  7. 

A/-,  thus  decrease  indefinitely  together,  they  be  all  multi- 

plietl  by  the  same  increasing  number,  ?;,  the  equimultiples  nAq, 

iiAr,  may  tend  to  finite  limits,  and  these  limits  are  defined 

to  be  the  simultaneous  differentials  of  the  related  quaternions  q. 
r,  s,  ,  and  are  written  dq,  dr,  ds, Simultaneous  differ- 
entials are,  therefore,  the  limits  of  equimultiples  of  simultaneous 
decreasing  differences.  If,  then,  in  A p  =  f{q-\- Aq)  —  f{q), 
while  the  finite  differences  Ap,  A  7  lie  indefinitely  decreased,  they 
be  multii)lied  by  a  number,  n,  ultimately  to  be  made  infinity, 
so  that 

nAi>  =  ,/[/(7  +  A7)-/(7)], 

and  we  pass  to  the  limit,  writing  dp  for  nAp,  and  dq  for  nAq^ 

we  have 

;  limit     r   /        d(i\  "I 

^'^'  =  .  =  x"[./\7+^j-/(ry)j 

a  formula  for  the  differential  of  a  single  explicit  function  of  a 
single  variable. 

IfQ=F(7,  r,  ), 

'^^^  =i  =  X  '^l^i'l  +  n-'dq,  r  +  n-'dr, )-F{q,  r )]    (108). 

In  these  formulae,  dq,  di\  are  auN'  assumed  variables,  no 

reference  having  been  made  to  their  magnitudes,  and  n  any 
positive  whole  number  conceived  so  as  to  tend  to  infinity.  To 
show  that  these  differentials  need  not  be  small,  as  also  the  ap- 
plication of  the  formula  to  the  differentiation  of  ordinary  scalar 
quantities,  let 

y  =  x-; 


w 

136  QUATERNIONS, 

then 

whence,  as  usual, 

A2/  =  2.i-Aa:  +  (Aa;)2, 

or,  n  being  a  positive  whole  number, 

n  A  y  =  '2xn  A  x  +  n'^{n  A  x)-. 

If,  now,  the  differences  A  y  and  A  x  tend  together  to  zero, 
while  71  increases  and  tends  to  infinit}'  in  such  a  manner  that 
iiAx  tends  to  some  finite  limit,  as  a,  we  have,  for  the  other 
equimultiple  n  A  y, 

11  Ay  =  2  xa  +  n~^  a-. 

But,  since  a,  and  therefore  cr,  is  finite,  n~^a-  tends  to  zero, 
and,  at  the  limit,  nAy  =  2xa.  Hence  the  limits  of  the  equi- 
multiples nAx  and  nAy  are  respective!}'  a  and  2x-a,  and 
clx  —  a,  dy  =  2xa  b}'  definition;    from  which 

cly  =  2  xclx. 

For  a  vector  function  we  should  write 

'^P'=n  =  l:''U{p  +  n-'dp)-fl,p)^     .      .     (169). 
and  for  a  scalar  function,  p  =  <f)(t), 

d,o=d[c^(0]  =  J'rSJc/>(^4-^)-c^(0l     .     (170), 
in  which  letter  t  and  dt  are  independent  and  ai'bitrary  scalars. 
64.    As  a  further  illustration  of  tlie  definition,  let 

p=cf>{t) 


APrLICATIONS    TO    LOCI.  137 

h1<.i'  =  P  (Fig.  r.l) 

Fi''.  04. 


be  the  equation  of  an}-  plane  cui-vc  in  ^[Mia 
a  vector  from  tlic  origin  to  a  \nni\t  v 
of  the  curve  ;  /  liring  any  arl)itrary  sca- 
lar representing  time,  for  example  :  so 
that  its  value,  for  any  other  point  i-'  of 
the  curve,  represents  the  interval 
elapsed  from  any  detinite  epoch  to  tin- 
time  when  the  point  generating  the 
curve  has  reached  v'.  *^ 

If  p'  be  the  vector  to  v'.  then 


is  strictly  the  finite  difference  between  p  and  p',  and,  if  the  corrc- 
.sponding  change  in  t  be  A^, 

ri''=  {p  +  Ap)-p  =  Ap=cj,{t  +  At)-  c/>(0  =  A  <^(0  ; 

where  op'=  cfy{t  -\-  A  t) ,  and  A  t  is  the  interval  from  p  to  p.' 

In^Ai,  p  Avould  have  reached  some  point  as  p",  for  which 
op"=  (f){t  -j--k  At),  on  the  supposition  that  pp"  is  described  in 
^At.  On  the  basis  of  this  closer  approximation  to  the  velocity 
at  p,  p  would  have  l)een  found  at  j/',  had  this  veloeity  remained 
unchanged,  such  that 

r//'=  ■2pp"=  2(op"-  op)  =  2[cf>{t  +  ^  A  0  -  <^(0]. 

For  a  closer  approximation  to  the  vector  descril)cd  in  A  t  with 
the  velocity  at  p,  suppose  at  the  end  of -^A  t  the  point  is  at  p'", 
for  which  op"'=  4>(t  +  ^A  0-  Under  this  supposition,  the  vec- 
tor described  in  A  t  Avould  have  been 

17/"=  3pp"'=  3(op"'-  op)  =  ;3[<^(^  +  ^A  0  -  </>(0]. 

and,  at  the  limit,  representing  the  niulti[)le  of  the  diminishing 
cliord  by  dp, 


138  QUATERNIONS. 

65.   Resuming  Equation  (167), 

dp  =  dj\q)  =  ^^n  [/(g  +  n-\lq)  -/(ry)],  («), 

the  second  member  may  be  written /(ry,  dq),  but  not,  as  ordi- 
XiiiY\\y,f{q)(lq. 

In  f(q,  dq),  dq  may  be  composed  of  parts,  as  q',  q'\  q"\ , 

with  reference  to  which /(g,  dq)=f{q,  q'+q"+  )  is  distrib- 
utive.    To  prove  this,  let 

dq  =  q'+q"-, 
we  are  to  prove  tliat 

f{q.  (/+(l")=f(fh  <l')+M^  <!")■ 

Since  before  passing  to  the  hmit,  tlie  second  member  of  (a) 
is  a  function  of  w,  q  and  dq,  we  may  express  tliis  function  liy 
the  symbol  /„(g,  dq),  and  write 

/(ry,  dq)  =  n[f{q  +  irhlq)  -f{q)^=f,Xq,  dq) , 
or 

f{q  +  n-'dq)  =f{q)  +  n-\f,Xq,  dq)  . 

Replacing  dq  by  q'  and  q"  in  succession,  we  liave 

f{q  +  n-'q')  =f{q)  +  n-'^q,  q') , 
f{q  +  n-'q")  =f{q)  +  n'^Mq,  q"), 

and,  following  the  same  law  of  derivation, 

f(q-\.n-'q"-\-n-'q')=f(q  +  n-'q")  +  n'\f,X(J  +  n-'q",  q'), 
'f{q  +  n-'q'+n-'  q")=f{q)  -\-  n'^f^iq,  q'+'l")  ^ 

from  wliich 

fn{q.  </+  q")  =.fn(q'  q")  +fniq  +  '^''q''^  q')^ 

the  Umiting  form  of  which,  for  ?i  =  x,  is 

f{q,q'-\-q")=f{q^q")+Aq^'/)     ■    ■   (I'l)- 


ArPLICATIONS   TO    LOCI.  lo'l 

wiiicli  !u:iy,  in  liko  inMiineT,  1)l'  cxU-iidcd  to  the  case  of 

dq=q'+q"+q"'+ • 

It  follows  from  the  above  that,  if  ;;=_/"(7,  .17/7), 

f{q.xdq)  =  xj\q,flq)       .      .      .      .      (172). 
If  Q  =  F(f/,  r, ),  whence,  Equation  (1G8), 

aq  =  aiF{q,r, )] 

=  l'^l,^^inq  +  'rhlq.r  +  u~Ulr, )-F{q.r )]. 

the  last  member  will  be  a  linear  and  homogeneous  function  of 

r?7,  (Jr ,  and  distributive   with   reference  to  each  of  them. 

Hence,  to  ditl'erentiate  such  a  function,  we  do  so  with  reference 
to  each  factor,  and  take  the  sum  of  the  results  obtained,  as  usual ; 
taking  care,  however,  not  to  make  use  of  the  commutative  prop- 
erty.    Thus  d{qr)  =  dq  .  r  +  qdr,  but  not  rdq  +  qdr. 

66.  When  q  \s  a.  function  of  any  variable  scalar  t,  represent- 
ing time,  for  example,  then,  if  t  be  given  a  finite  increment  A  t, 
for  which  the  corresponding  one  of  ^y  is  A  q,  we  have 

A  7  =  A  io  +  A  .vi  +  A  )/j  -\-Azk; 

and.  if  the  several  parts  of  the  (luaternion  varv  continuously 
with  the  independent  variable  t,  at  the  limit  we  ma\'  form,  as 
usual,  the  differential  coefflcient 


dt       dt       dt        df       df'' 

The  successive  dilTerential  coeflicients,  as  also  the  partial  ones, 

when  q  =  <f)(t,  v, ),  are  derived  from  the  quadrinomial  form  in 

the  same  manner. 


1-10  QUATERNIONS. 

67.  Examples. 

1 .    To  find  dlq. 


clTq  _  rlNir-  +  ■^"  +  if  +  ^' 
(It   ~  clt 

1   /    (ho  ,     (Iv  ,      (hi  ,     (h\ 
T(j\     (It  at         (It        dtj 


dq 


f?Tg 

■.H± 

Ijq 

2.    (TpY- 

=  -p\ 

The  first  member  being  a 

scalar,  wt 

have 

2Tpc?T|0. 

From  the 

second  membei 

rf(p') = 

_  limit 
"  n  =  X 

;[(p  +  »- 

dp)-- 

=  limit  pf?p  +  f/p  .  p  +  )i  \dpy 
=  pdp  +  dp  .  p  =  2  Spr?p. 
Equating 

TpdTp  =  —  Spdp. 

From  this  we  may  obtain 

f?Tp  =  -S.Upf/p  =  S^, 

Tp       '    p' 
3.    To  find  cWq.     Wc  liave 

(?Tr/  .  Dry  +  dVq  .  IVy  =  dq, 


APPLICATIONS    TO    LOCI.  141 

whence 


clTq  .  Vq      dV(j  .  Tq  _  dq 
TqVq  TqVq     ~  q  ' 


(Wq 

Vq 

dTq 
Tq' 

and.  substituting 

from  Ex.  2. 

(Wq 
Vq 

-.'h- 

7 

'1 

or 

dVq 

Vq 

dVq  = 

.v^ 

.  Ury. 

(1 

4.    From  the  above  expressions  for  dTq  and  f?Ug,  we  have 
dq  =  dTq  .  Vq  +  TqdVq 

Vq  Vq) 


(^s|  +  v|)g 


as  the  form  nnder  Avhich  the  diftbrential  of  a  quaternion  may 
always  be  written. 

5."  To  find  dVp.     AVe  have,  from  p  =  TpUp, 

dp  =  dTp  .  Up  +  TpdUp, 


from  Ex.  2, 


dp      dTp      dVf> 
P        Tp        Up 
_  gdp  _^  dVp 
P        Up' 

fro 

dVp 
Up 

'?,o       ^  dp  _  -,  dp  _  „  pdp  _ 
P            P            P             P' 

p" 

vhence,  also, 

dUp  =  '^^'-^'^^^ 
{Tpf 

142  QUATERNIONS. 

6.  From  the  above  expressions  for  dTp  and  cWp, 

7.  That  S,  V  and  K  are  commutative  witli  d  is  seen  from  the 
following : 

q  =  Sq  +  Vry, 
whence 

dq  =  cZSry  +  c7Vg,  (a) 

and,  since  dq  is  a  quaternion, 

dq  =  Hdq  +  Ydq,  (6) 

hence 

dSq  =  Mq     and     d\q  =  Ydq.  (c) 

Again 

Kg  =  Sry  -  Vg, 
whence 

(■?K(/  =  dSq  -  dJq, 

and,  taking  the  conjugate  of  dq  in  either  (b)  or  (a),  we  have, 
with  or  without  (c), 

dKq  =  JLdq. 

8.  {Tqy-==q1Lq. 

2  TqdTq  -  j'^^J,  v  [(ry  +  irV/g)  (Kg  +  n-hlKq)  -  ryKg] 

=  limit  [dq{Kq  +  n'^Kdq)  +  gKdg] 

=  c7g.K7  +  gKrfg 

=  K  .  qKdq  +  f/KfZg 

=  2  S  ,  fyKfZ(/  =  2  S  .  Kqdq,  [Equation  (80)] 

or,  since  Tg  =  TKg  and  UKg  =  U-  =  — , 

g      Ug 

1 

fZTg  =  S  .  U  - dg  =  S  .  Ug-'  dg. 

If  g  =  a  vector,  as  p,  then,  since  Ko  =  —  p,  this  becomes 

dTp  =  -  S  .  U/:;dp, 
as  in  Ex,  2. 


APPLICATIONS    TO    LOCI.  143 

9.  r=(p. 

=  limit  [f/r?7  -\-(lq  .  q  +  u-^chj)-] 
.-.  dr  =  2  Sqdfj  +  2 SqXd'j  +  2 SdqXq. 
If  (y  =  a  vector,  as  p,  then  Sr/  =  U.  Sf/7  —  0,  and 
(7(p-)=2S,of7p 
as  in  Ex.  2. 

10.  ?■=  V7.     Then  q  =  r-,  and,  as  before, 

f^y  =  rc/>'  +  dr  .  ?•. 

Operating  with  rx  and  xK/%  in  succession, 

rdq  =  /•-(//■  +  >■(?'■  •  '>', 
dq  .  Kr  =  rdr  .  Kr  +  f/r  .  rKr 
=  rf?r  .  K/-  +  (Tr)-(/r, 
or,  adding, 

rdq  +  r?(7  .  Kr  =  [r  +  (T/' )']'//•  +  *•(«'•(?•  +  Kr) 


which  gives  r^r  =  d\/q  in  terms  of  d^. 


11.    qq~'  = 

=  1. 

AV 

'e  ': 

have 

qo 

Hq- 

')  +  dq 

•7" 

1  = 

0. 

Operating 

with 

T 

'  X 

7"' 

'7' 

(i'l 

dl 
'I 

=  -'-d 
7 

''7 
7  • 

•  7" 
1 

7 

1  _ 

:0. 

144  QUATEllNIONS. 

If  q  =  a  vector,  as  p, 

,11,1 
a  -  =: dp- 

P  P      P 

1,1,1,       11, 

= dp-  -\-  —dp -dp 

p      p      p-  p  p 

dp      1/1  ,,    ,       1\ 

P'      P  \P  PJ 

p'    p'  p 

P\P  Pj  P  P 

12.  Differentiate  ^Vq. 

rfSUg  =  MVq  =  S  .  V^Ug       [Exs.  7  and  3.] 
dq 

'1 

q\}\q  ^ 

13.  Differentiate  VUg. 

dq 
dVUr/  =  V  .  dVq  =  V  .  V  y^  Ug        [ Plxs .  7  and  3 .  ] 

=  V.Ug-M^(f?g  .  (y-'). 

14.  Differentiate  TVUg. 

dXVUg^S^M  [Ex.2.] 

f?g 

~^UVg  UVg 

r?g 


APPLICATIONS    TO    LOCI.  145 


The  Right  Line. 


As  ill  Cartesian  coordinates,  the  form  of  the  equations  of  a 
right  line,  as  of  other  loci,  will  depend  upon  the  assumed  con- 
stants, and  in  an}-  given  problem  one  form  may  be  more  con- 
venienth'  used  than  another. 

68.  Right  line  through  the  origin. 

If  o  be  the  initial  point,  or  origin,  and  p  =  or  a  variable  vec- 
tor in  the  prolongation  of  a  =  oa,  then 

p  =  Xa (1'^'^) 

is  the  equation  of  a  right  line  through  the  origin  in  the  direction 
of  the  constant  vector  a. 
The  equations 

UP  =  ^'^1 (174) 

Vap  =  0      j 

obviously  refer  to  the  same  right  line. 

Since  an}-  line,  represented  as  a  vector  l)y  a,  is  parallel  to 
p=xa,  we  may  say  that  the  above  equations  are  those  of  a  right 
line  through  the  origin  parallel  to  a  given  line  ;  or,  a  being  a 
point  given  by  a  =  oa,  they  are  the  equations  of  a  right  lino 
through  the  origin  and  a  given  point. 

69.  Parallel  lines. 

If /3  =  OB  be  a  constant  vector  to  a  given  point  n,  then 

P  =  /8+.^-a (IT-'') 

is  the  equation  of  a  right  line  through  a  given  point,  and  parallel 
to  a  given  line,  as  p'=  xa  through  the  origin.  Or,  a  being  a  given 
vector,  it  is  the  equation  of  a  right  line  through  a  given  point 
and  having  a  given  direction.  If  a  is  an  undetermined  vector, 
it  becomes  the  general  equation  of  any  one  of  the  infinite  num- 
ber of  right  lines  which  may  be  drawn  through  a  given  point.  If 
0  and  li  coincide,  /3  =  0,  and,  as  before,  p  =  xa. 


146 


QUATERNIONS. 


a  remaining  the  same,  and  (3'  =  ob'  being  a  vector  to  an}-  other 
point  b'  for  the  equations  of  two  parallels,  we  have 


or,  since  a  and  p  —  y8  are  parallel, 

Va(p-y8)  =  0 
Va(p-^')=0 


(176), 


(177). 


70.   Right  line  through  two  given  points. 

If  OA  =  a    (Fig.    Go),  OM  =  (i   are   the  vectors  to  the  given 
points,  and  p  the  variable  vector  to  anj- 
Fig  65  point  R  of  the  line  whose  equation  is  re- 

quired, we  have 


and 


AR  =  XXB.  =  .r(/3  —  a), 
OR  =  OA  +  AR, 

or,  for  the  required  equation, 

p  =  a  +  x{P-a)        .      (178), 

which,  if  one  of  the  points,  as  a,  coincides  with  the  origin, 
becomes  p  =  xj3,  as  before. 

We  have  seen.  Art.  55,  that  if  Sa/?y  =  0,  a,  /S  and  y  are  com- 
planar.     Replacing  y  b}'  the  variable  A'ector  p, 

Sa^p=0 (179) 

is  the  equation  of  a  plane^  since  it  expresses  the  condition  that  p 
is  complanar  with  a  and  fi.  If  we  have  also  Sayp  =  0,  the  two 
equations,  taken  together,  represent  the  line  of  intersection  of 
these  two  planes. 

These  equations  may  be  obtained  from  the  line  p  —  xa  by  ope- 
rating with  S(Va/?)  X  and  S(Vay)x  ;  or,  conversely,  to  find  the 
equation  of  the  line  in  terms  of  known  (luantities,  having  given 


SaySp  =  0,      Sayp  =  0, 


APPLICATIONS   TO   LOCI.  147 

write  these  latter  under  the  form 

S.pVa/?  =  0,       S.pV.xy=0, 

whence  it  appears  tliat  p  is  perpeiKheiilar  to  both  Vaj8  and  Vay, 
and  is  consequently  parallel  to  the  axis  of  their  product ; 
therefore 

p  =  y\  .  \a/3\ay 

=  1/  ( ySa^a  -  aSaySy )  [Eq .  ( 11 2  )  ] 

=  —  yaSa(3y, 


or,  putting  —  ?/Sa/Jy  =  x. 


p  =  Xa. 


71.   Right  line  perpendicular  to  a  given  line. 

1.  Let8  =  OD  (Fig.  GO)  be  a  vector  through  the  origin.  To 
find  the  equation  of  dc  through  its  extremity  ^j^.  gg 

and  perpendicular  to  it.      Now  p  —  8  is  a        u  r      c 

vector  along  dk,  and  therefore  l>y  condition 

S8(/3-8)  =  0. 

Whence  SSp  =  -(T8)%  or 

S^p  =  c,  a  constant (180). 

In  oi'der  that  p,  p  —  8  and  8  be  comi)lanar,  we  must  have 

S  .  8p(/3-8)  =  0, 
or 

S.(\8p)(p-8)  =  0. 

2.  p  —  8,  being  perpendicular  to  both  8  and  V8p,  will  be 
parallel  to  the  axis  of  their  product,  or  to  V  .  S\8p.  Hence,  if 
y  =  oc  be  a  vector  to  any  point  c,  in  the  plane  of  on  and  dk,  the 
equation  of  a  right  line  through  a  given  point  c.  perpenrlicular  to 
a  given  line  od,  will  ]yc 


p  =  y  ■+  .xV  .  8V8y  . 


(181). 


148  QUATERNIONS. 

3.  If  the  perpendicular  is  to  pass  through  the  origin,  then, 
from  Equation  (180), 

SSp  =  0 (182), 

or,  in  anotlier  form,  from  Equation  (181),  y  being  parallel  to 
V  .  8V5y, 

p  =  ?/V.SV8y (183). 

4.  The  student  will  find  it  useful  to  translate  the  Quaternion 
into  the  Cartesian  forms.   Thus,  from  Equation  (180) ,  if  rod=6, 

SSp  =  -T8Tpcosd, 

whence,  if  r  and  d  represent  the  tensors, 

rd  cos  6  =  d-,     or    r  =  — — , 
cos  6 

the  polar  equation  of  a  right  line. 

5.  Equation  (181),  of  a  line  through  a  given  point  and  per- 
pendicular to  a  given  line  through  the  origin,  may  be  otherwise 
obtained,  as  follows : 

Let  y  and  8,  as  before,  be  vectors  to  the  point  and  along  the 
given  line,  respectively-,  and  f3  a  vector  along  the  required  per- 
pendicular, whose  equation  will  then  be 

p  =  y  +  xp.  (a) 

To  eliminate  ^  we  have  the  conditions 

SS/?  =  0, 

since  8  and  /3  are  perpendicular  to  each  other,  and 

Sy8/3  =  0, 

since  y,  8  and  (3  are  complanar.  But  \8y  is  perpendicular  to  this 
plane,  and  therefore  V  .  SV8y  is  parallel  to  /? ;  hence,  substitut- 
ing in  (a), 

p  =  y  -f  x\  ,  8\8y, 
or  simply 

p  =  y  +  xSV8y. 


APPLICATIONS   TO   LOCI.  149 

JfSy^fS^O,  y,  ^  iind  /3  are  not  coniplanav,  and  the  problem  is 
indeteniiinate  ;  wliieh  also  api)ears  froui  ((f),  by  operating  with 
X  S  .  8,  whence,  since  S/?S  =  U, 

Sp8  =  SyS, 

a  result  which  is  independent  of  /?,  and  an  infniite  nuni])er  of 
lines  satisfy'  the  condition. 

6.  If  the  line  to  which  the  perpendicular  is  drawn  does  not 
pass  through  the  origin,  let 

p  =  (3  -hxa  (a) 

be  its  equation.     Then,  if  p  be  the  vector  to  the  foot  of  the  per- 
pendicular, we  have  Sa(/3  —  y)  =  0,  or 

Sa(a;a  +  /3-y)  =  0,  (6) 

because  the  line  is  perpendicular  to  (a) ,  or  its  i)arallel  a.    Hence, 
from  (&), 

a;a  =  a-'Sa(y  —  (3), 

or,  fbr  the  perpendicular  p  —  y, 

^_y  =  a-a  +  ^-y  =  a-^Sa(y-/5)-a-'a(y-^) 
=  _«-iVa(y-/3). 

Its  length  is  evidentl}- 

TV[Ua.(y-/3)] (184). 

7.  This  perpendicular  is  the  shortest  distance  from  the  point 
to  the  line.  The  problem  ma}-,  therefore,  be  stated  thus  :  to 
find  the  shortest  distance  from  c  to  the  line  p  =  xa  -{-  (S.  p  being 
the  vector  to  the  foot  of  a  line  from  c  to  any  point  of  the  given 
line,  this  vector  is 

fi  +  Xa  —  y, 

and,  in  order  that  its  length  be  a  mininunn, 

dT(^  +  a;a-y)  =  0 
=  T{f3  +  Xa-y)dT{P  +  Xa-y) 
=  —  S[(/3  -f  xa  —  y)a'\dx  =  0, 


150  QUATERNIONS. 

or 

^(f3-i-Xa-y)a  =  0, 

that  is,  the  Ihie  must  be  perpendicular  to  p  =  xa-\-  (3. 

8.  If  the  perpendicular  distance  from  the  origin  to  p  =  (3  -\-  xx 
is  required,  p,  being  as  before  the  vector  to  the  foot  of  tlie  per- 
pendicular, coincides  with  it ;  hence,  y  being  zero,  and  8  repre- 
senting this  value  of  p, 

S  =  xa  -\-  (3. 


Operat 

ing 

with 

X 

S 

.  8,  since 

SaS  =  0, 

Hence 

-(Tsy 

TS 

=  S;88. 

_  S  .  yQTSUS 
TS 

.  /3US  .     . 

.     .     (185). 

72.  "We  are  to  observe  that  the  foregoing  equations  of  a  right 
line  are,  as  remarked  in  Art.  60,  all  linear  functions  involving, 
explicitly  or  implicitlv,  a  single  real  and  independent  variable 
scalar.     Such  is  evidently  the  case  for  such  equations  as 

p  =  xa,  [Eq.  (173)1 

p  =  ^  +  xa,  [P:q.  (175)] 

p  =  a  +  x((S-a).  [Eq.  (178)] 

So  also  for  the  implicit  forms,  as  \ap  =  0  [Eq.  (174)]  ;  em- 
ploying the  trinomial  forms 

a  =  a>+hj  +  ck, 
p  =  xi  +  ijj  4-  zk, 
we  have 

ap  =  (bz  -  cy)  i  -t-  (ex  -  az)j  +  {ay  -  hx) k  -  (ax  +  by  +  cz) . 

Whence 

\ap  =  (bz  —  cy)  i  -f-  (ex  —  az)j  -j-  (ay  —  bx) k  =  0  ; 
.-.  bz  —  ey,     ex  =  az,     ay=bx, 

in  which  x  and  y  are  functions  of  z. 


ArPlJCATloN.S    TO    LOCI. 


151 


The  Plane. 
73.   Equation  of  a  x)hn\e. 

1.  If,  in  the  equation  S  ,  8/3  =  0,  which  denotes  that  /?  is  per- 
pendicular to  8,  we  replace  /3  by  the  varial)le  vector  p, 

s.8/j  =  o (i.sn) 

is  the  equation  of  a  plane  through  the  origin  perpendicular  to  8. 

2.  Or,  let  8  =  CD  (Fig.  Gf!)  be  the  vector-  Fv^.m  {bis). 
perpendicular  on  the  plane,  and  dr  an}-  line       d  no 
of  the  plane. 

Then 

S3(p-8)  =  0. 
S3p  =  8-  =  -(TS)S 
or 

S^p  =  c\  a  constant       .     .      (1S7) 

is  the  general  equation  of  a  plane  perpendicular  to  8.     Here  dk 
is  any  line  of  tlie  plane  ;  and,  if  y8p  —  e. 


Sip  =  an  indeterminate  quantity 


(188), 


If  tlie  plane  pass  through  the  origin,  we  have,  as  before, 
SSp  =  0.  Conversely,  if  S3p  =  c  is  the  equation  of  a  plane,  8  is  a 
vector  perpendicular  to  the  plane. 

3.  The  equation  of  a  plane  through  tlie  origin  i)erpendicular 
to  8  may  also  be  written  in  terms  of  an}-  two  of  its  vectors,  as 
y  and  (3 ; 

p  =  -^-p  +  yy- 

Both  of  these  indeterminate  vectors  may  be  eliminated  by 
operating  witli  S  .  8  x  ,  whence 


SSp  =  0 

as  before  ;  or  one  may  lie  eliminated  by  operating  with  V  .  /S  X  , 
whence 

y(3p  =  zs, 


152  QUATEKNIONS. 

from   which   we   may    again  derive    S?>p  =  0  b}'  operating  with 
V  .  8  X  ;  for 

V  .  8V/Sp  =  \zS-  =  0 

=  pSS^-;8S3p,  [Eq.  (Ill)] 

whence,  since  SS/S  =  0,  S5p  =  0. 

4.  The  equation  of  a  plane  through  a  point  n,  for  which 
OB  =  /3,  and  perpendicular  to  8,  is 

SS(p-^)  =  0 (ISO). 

o.  Having  the  equation  of  a  plane,  SS,o  =  c,  to  find  its  dis- 
tance from  the  origin,  or  the  length  of  p  when  it  coincides  with 
8,  we  have  p  =  .^•3  ;  hence 

SSp  =  c  =  S.x-5-  =  xh-, 
or 

which,  in  p  =  xS,  gives 


c 

g2' 


c 
^  =  8' 
or 

Tp  =  4 (190). 

74.  To  find  the  equation  of  a  plane  through  the  origin,  making 
equal  angles  ivith  three  given  lines. 

Let  a,  /3,  y  be  unit  vectors  along  the  lines.  The  equation  of 
the  plane  will  be  of  the  form 

SS/j  =  0. 

By   condition,    Sa8  =  S^8  =  SyS  =  TSsiuc/)  =  x-,    4>   being   the 
common  angle  made  by  the  lines  with  the  plane. 
Hence 

Hindi  =  ^- 
^  ■   T8 


APl'LICATIONS    TO    LOCI.  153 

To  elimiiuvto  8,  wc  liiivc.  ihnn  VA\nn\\on  (118), 

SSa/?y  =  Va/SSyS  +  T/iyS.xS  +  YyaSySS, 

ami.  by  condition, 

8Sa^y  =  .7-(  Va/3  +  V^y  +  Yya)  . 

Tlio  vector  rcprosented  by  the  parentliesis  is,  then,  the  pei'- 
pendiciilar  on  the  plane,  whose  eqnation,  therefore,  is 

Sfj(\a/3  +  V/3y  +  Yya)  =  0  .      .      .      .     (191 ) . 

and  the  sine  of  the  angle  cfi  is 

Sa^y 

T(Va;8+V/8y  +  Vya)* 

75.   Equation  of  a  plane  through  three  given  points. 

Let  a,  /?,  y  be  vectors  to  the  given  i)oints  ;  then  are  the  lines 
joining  these  points,  as  (a  —  ^g),  (/5  —  y),  lines  of  the  plane.  If 
p  is  the  variable  vector  to  anv  point  of  the  plane,  p  —  a  is  also  a 
line  of  the  plane.     Hence 

S(p-a)(a-/i)(/i-y)=(). 

or 

S(pa;8  -  pay  -  pfi'  +  pfty  -  a'ft  +  u'y  +  ayS"  -  aySy)  =  0. 

But 

S(-p^-)  =  0,  S(-a-/8)=:0,  etc., 

S(  —  pay)  =  Spya  =  S  .  pVya, 
Spa^=S  .  p\a/3.  etc., 
hence 

S.p(Vr;/3  +  V/5y  +  Vyu)-Sa^y=0     .       .       (1!I2). 

which,  by  making  tlie  vector-parenthesis  =  8.  may  be  written 
under  the  form 

Sp8  -  S:i/3y  =  0, 


154  QUATERNION'S. 

in  which  S  is  along  the  perpendicular  from  the  origin  on  the 
plane.  When  p  coincides  with  this  perpendicular,  p  =  .t3,  and, 
from  the  above  equation, 

xh-  =  Sa^y, 

or,  for  the  vector-perpendicular, 

Sa/3y 


p  =  a;S  =  8  ^  Sa/5y 


Yajd  +  \(3y  +  Vya 


76.  We  observe  again,  from  inspection  of  tlie  equations  of  a 
plane,  that,  as  remarked  in  Art.  CO,  they  are  linear  and  func- 
tions of  two  indeterminate  scalars.     Thus,  for  a. plane  through 

the  origin 

SSp  =  0,  [P]q.  (ISG)] 

employing  the  trinomial  forms  S=ai  +  bJ+cl-  and  p=xi  +  )iJ-\-zk, 
we  obtain 

8p  =  {hz  -  cy)  I  +  {ex  -  az)j  +  {ay  -  hx)k  -  {ax  +  by  +  cz) , 

the  last  term  of  which  is  the  scalar  part ;  hence 

ax-\-ljy  +  cz  —  Q^ 

the  equation  of  a  plane  through  the  origin  o,  perpendicular  to  a 
line  from  o  to  (a,  h,  c),  which  may  be  written /(a;,  ?/,  z)=^Q; 
or  as  a  function  of  two  iudeterminates.  In  the  same  way,  from 
an  inspection  of  the  other  forms, 

p  =  xa  +  y(i,  .  [Art.  73,  n] 

p  =  Z  +  xa  +  7//3, 
SSp  -  c'  =  ax  +  %  +  r^  -  c'  =  0 ,  [Eq.  ( 1 87 )  ] 

we  observe  they  are  linear  functions  of  two  indeterminate  scalars. 

77.  Exercises  and  Problems  on  the  Right  Line  and 
Plane. 

1.  fi  and  y  being  vectors  along  two  given  lines  luhich  intersect 
at  the  point  a,  to  which  the  vector  is  oa  =  a,  to  write  the  equation 
of  a  line  perpendicular  to  each  of  the  two  given  lines  at  their 
intersection. 


APPLICATIONS    TO    LOCI.  loo 

\l3/  is  a,  vector  m  the  direction  of  the  required  line,  whose 
eqiuition.  therefore,  is 

P  =  a+x\f3Y (1!»;5). 

If  a'  =  oa'  be  a  vector  to  any  other  [)oint,  us  a,'  tlien  is 

^  =  a'  +  x\(3y 

the  ('(juation  of  a  hne  tlu'ough  a  given  point  perpendicuhir  to  a 
given  plane  ;  the  latter  being  given  by  two  of  its  lines. 

2.  a  (Did  (3  being  vectors  to  two  (jiven  jtoinfs,  a  aitd  n,  and 
S5,Q  =  c  the  equatioit.  of  a  given  plane,  to  find  the  equation  of  a 
plane  through  a  and  h  perpendicalar  to  the  given  plane. 

?t,  p  —  a  and  a  —  [3  are  lines  of  the  required  plane,  hence 

S(p-a)(a-^)8  =  0, 

or 

S/)(a-/?)S  +  Stt/38  =  0 (1"J4) 

is  the  required  equation. 

3.  oc  =  y  l>('i)ig  a  vector  to  a  given  point  c,  and  p  =  a  +  xf3, 
p  =  a'-\-  x'(3'  the  equations  of  tivo  given  lines,  to  write  the  equation 
of  a  plane  through  c  parallel  to  the  two  given  lines. 

If  lines  be  drawn  through  the  given  point  parallel  to  the  given 
lines,  thej  will  lie  in  the  required  plane.  As  vectors.  (3  and  (3' 
are  such  lines,  and  p  —  y  is  also  a  line  of  the  plane.     Hence 

S/3/5'(,o-y)=0 (195) 

is  the  required  equation.  If  y  =  a,  or  a',  it  is  the  equation  of  a 
plane  through  one  line  parallel  to  the  other.  Or.  if  y  is  inde- 
terminate, the  general  equation  of  a  plane  ^xirallel  to  two  given 
lines. 

Otherwise  :  the  equation  of  a  i)lane  through  the  extremity  of 
y  parallel  to  two  given  lines,  whose  directions  are  given  b^- 
a  and  /?,  is  evident!}'  p  =  y  +  xa  +  y(3. 

4.  To  find  the  distance  between  two  points, 
a  and  /?  l)eing  vectors  to  the  [)()ints, 

y  =  /3-a. 
Squaring 

<r  —  Ir  -\-  a'-  —  2  ab  cos  c. 


156  QUATERNIONS. 

5.    A  plane  being  r/iven  by  two  of  its  lines,  ft  and  y,  to  write 
the  equation  of  a  right  line  through  a  perpendicular  to  the  plane. 
Let  OA  =  a.      Draw  two  lines  through  a  parallel  to  yS  and  y. 
Then 

p  =  a  +  x\(iy (19G). 

If  the  plane  is  given  by  the  equation  SSp  =  c,  then 

p  =  a  +  .rS (197). 

G.  Fi)id  the  length  of  the  perpendicular  from  a  to  the  plane., 
in  the  jyreceding  example. 

Operating  on  Equation  (197)  with  S  .  8  X 

SSp  =  SSa  +  x'S-  =  c, 
or 

a;82  =  c-SSa; 
.-.  xB  =S-^(c-S,^a) (198). 

7.  S8(p  — )8)  =  0,  Equation  (189),  being  the  equation  of  a 
plane  through  v,  perpendicular  to  8,  to  Jind  the  distance  from  a 
pjolnt  c  to  the  plane. 

Let  y  =  oc.  The  perpendicular  on  tlie  plane  from  c,  being 
parallel  to  8,  will  have  for  its  equation 

p  =  y  +  a;8. 

To  find  X-,  operate  with  S  .  8  x  ,  whence 

SSp  =  SSy  +  .1-5-, 

or,  from  the  equation  of  the  plane, 

SSy  +  xS^^SS/?; 
.-.  a;S  =  -S-^S3(y-^), 
and 

.'rTS  =  TS-iSS(y-iS)  =  S[U8  .  (y-^)]. 

8.  Write  the  equation  of  a  plane  through  the  parallels 

p  =  a+  x(3, 
p  =  a'+  X^. 


APPLICATIONS    TO    LOCI.  157 

9.  Write  the  equation  of  a  ■plane  throiKjk  the  line 

perpendicular  to  the  plane 

SSp  =  0. 

10.  Given  the  direction  of  a  vector-perpendicular  to  a  plane, 
to  find  its  length  so  that  the  plane  maij  meet  three  given  planes  in 
a  point. 

Let  8  be  the  given  vector-porpcndiciilar,  and 

Sa/3  =  rt,      S/3p  =  ^,      Syp  =  c 

the  eqnations  of  the  given  planes.  If  tlie  eqnation  of  the  plane 
be  written 

SSp  =  X, 

then  X  must  have  such  a  value  that  one  value  of  p  shall  satisfv 
the  equations  of  all  four  of  the  planes.  From  Equation  (118) 
we  have 

pSa/3y  =  Va/3Syp  +  V/3ySap  +  VyaS^p 
=  c\a(S  +  a\(3y  +  b\ya. 

Operating  with  S  .  8  X  ,  to  introduce  .r, 

X^af3y  =  cSBa(3  +  aSS/?y  +  hS^ya. 

11.  To  find  the  shortest  distance  between  tioo  given  right  lines. 
Let  the  lines  be  given  by  the  equations 

p  =  a  +  xp,  (a) 

p  =  a'  +  .r'/5:  {h) 

The  equation  of  a  plane  through  either  line,  as  (/>),  parallel  to 
the  other  («),  is  [Equation  (195)] 

Sy8/?'(p-a')  =  0.  (r) 

V/5/3'  is  a  vector-perpendicular  to  tliis  plane.  Hence,  if  >/\fi(3' 
be  the  shortest  vector  distance  between  the  lines,  we  have,  since 
'I  —  a'—  yYfS^'  is  a  vector  comi)lanar  with  /3  and  fS', 

S(3/3'{a-a'-l/\/3/3')^0, 


158  QUATERNIONS. 

or 

S(S/3^'  +  V^^')  (a  -  a'  -  7/V/3/5')  =  0, 
whence 

or,  dividing  by  T(Y^/3'), 

T(?/V^^')=TS[(a-a')lI(V^^')].       .       .    (I'J'J), 

the  s^'mbol  T  denoting  tliat  only  the  positive  niunerieal  viihie  of 
the  scalar  is  taken. 

Otherwise :  since  the  distance  is  to  be  a  niiiiimum, 

dT{p'-p)  =  0, 
whence 

^{p'-p){l3'dx'-(^dx)  =  0, 
or 

S(p'-p)^  =  0     and     S(p'-p)y8' =  0, 

or  the  shortest  distance  is  their  common  perpendicular,  whose 
length  may  be  found  as  above. 

12.    Given  SSj  p  =  d^  and  SS^p  =  do,  the  equations  of  two  planes, 
to  find  the  equation  of  their  line  of  intersection. 
This  equation  will  be  of  the  form 

p  =  rnSi  +  »  So  +  .rVSi  8,.  ( (I ) 

To  find  m  and  ?(,  we  have,  from  (a), 

SS,p  =  m.h-1  +  j^SSjSo, 
SSop  =  wSj-  +  ?iiSSi82i 

from  which  we  olitain 

_  SS,p-7^SSi8..  _  SSop  -  nl? . 
*''~  8f  "       SSiSo       ' 

S8iS«S5ip  —  Sj'SSo/o 


But 


—     (88182)^-81^8/ 
(S8i82)--8,-8/  =  (Y8,82)-; 


APPLICATIONS    TO    LOCI.  150 

And  siinihirly 

Substituting  these  values  in  {<() 

(h  SS,  82  -  d,  8," ,       (1,  S5, 8,  -  (1, 8,; 


P^ 


8.-  +  ^    '. :  ^.  ^.,    -  8,  +  .ry8, 8,„ 


(V8i8.)^        -  (V6,8,)- 


Avliicli  is  the  equation  of  the  retiuired  line,  a  less  useful  form  than 
those  of  the  two  simple  conditions  of  Art.  70. 

If  the  two  planes  pass  through  the  origin,  then  also  does  their 
line  of  intersection  ;  and  since  every  line  in  (me  plane  is  [)erpen- 
dicular  to  81,  and  every  line  in  the  otiier  to  80.  V8, 80  is  a  line 
along  the  intersection,  as  in  (a),  and  the  e({uation  becomes 

p  =  .r\8^8, (200). 

13.    The  planes  beiiirj  tj'nx'u  as  in  Equation  (189), 

S8(p-/S)  =  0,  (a) 

SB'{p-fS')  =  0,  {b) 

to  find  the  line  of  intersection. 

The  vector  p  to  an}'  point  of  the  hue  must  satisly  lioth  {a) 
and  (/>).  This  vector  may  be  decomposed  into  three  vectors 
l)arallcl  to  8,  8'  and  yh^l  which  are  given,  and  not  complanar, 
hy  Equation  (118) ;  whence 

pS  .  88'V8S'  =  Sp8V(8' .  n?>')  +  SpS'V(V88' .  8)  +  S(pV88') V8S', 

or,  from  (a)  and  (/j), 

-  p(TV88')-  =  S5/3y(S' .  VS8'')  +  SS'/?'V(V88' .  8)  +  SSS'pV88'. 

or,  since  SSS'p  is  the  only  indeterminate  scalar,  putting  it  eciual 
to  X",  we  have 

-  p(TV88')2  =  S8y8V(8' .  Y8S')  +  SS'/3'V(y88' .  8)  -f  .rV8S; 

If  the  planes  pass  through  the  origin,  in  which  case  jB  and  (i' 
are  zero,  we  have,  as  before, 

p  =  xy8h'. 


160  QUATERNIONS. 

14.    To  lorite  the  equation  of  a  plane  through  the  origin  and 
the  line  of  intersection  of 

SS(p-/?)  =  0,  («) 

If  p  is  such  that  SSp  =  SS/3,  then  also  SS'p  =  SS'/S;  and  both  the 
above  equations  will  be  satisfied.     Hence,  from  (a)  and  {b) 

S3pSS';Q'-S5/5SS'p  =  0, 

which  is  also  a  plane  through  the  origin.     This  equation  may 
also  be  written 

S[(8SS'/3'-S'S3y8)p]=:0, 
which  shows  that 

8S5'/3'-8'S3/? 

is  a  vector-perpendicular  to  the  plane,  and  therefore  to  the  line 
of  intersection  of  (a)  and  (&). 

1.3.    To  find  the  equation  of  condition  that  four  jmnts  lie  in 
a  7^^ane. 

If  the  vectors  to  the  four  points  be  a,  (3,  y,  8,  then,  to  meet 
the  condition, 

8  —  a,      8-/3,      8  —  y 

must  be  complanar,  and  therefore 

S(S-a)(S-^)(S-y)=0, 

whence 

SS;8y  +  SaSy  +  Sa/3S  =  Sa/3y        .       .       .      (201), 

which  is  the  equation  of  condition. 

Or,  X  and  y  being  indeterminate,  we  have  also 

8  =  a+.X-(^-a)  +  ,v(y-/3), 

or 

8  -f  (a;  -  l)a  +  (^  -  x)/3  -  uy  =  0, 
and 

l  +  (x--l)  +  (^-x)-^  =  0. 


APPLICATIONS    TO    LOCI.  IGl 

Or,  in  £?oneral. 


(202), 


aa  +  hfS  +  Cy  -\- (li)  =  {}  ) 

<'  +  b  +  c  +  (/  =  ())■ 

are  the  sufficient  coiulitions  of  eoniplaiuirity. 

Tliose  conditions  are  analogous  to  Equation  (D). 

16.  Given  the  three  planes  of  a  triedral,  to  Jinil  the  equations 
of  2^lf(nes  tJirongh  the  edges  perpendicular  to  the  opposite  faces^- 
and  to  show  that  they  intersect  in  a  right  line. 

Taking  the  vertex  as  the  initial  point,  let 

Sap  =  0,  (a) 

S/?p  =  0,  (/.) 

Syp  =  0  (c) 

be  the  equations  of  the  plane  faces.  Then  Vu/3  is  a  vector  par- 
allel to  the  intersection  of  (a)  and  {!>),  and  V  .  yVa/3  is  a  vector 
perpendicular  to  the  required  jjlane  through  their  common  edge. 
Hence  the  e(][uation  of  this  plane  is 

S(pV.yy«/5)  =  0.  (a') 

Similarly,  or  bv  a  cyclic  change  of  vectors, 

Sip\.a\/3y)  =  0,  (b') 

S(pV./i\»=0  (c') 

arc  the  equations  of  the  other  two  planes. 

If  from  their  common  i)oint  of  intersection  normals  arc  drawn 
to  the  i)lanes,  then  are  V  .  yVa/5,  V  .  aY/3y  and  Y  .  /JVya  vector 
lines  parallel  to  them  ;  but,  Equation  (123), 

V(yV«^  +  «V/3y  +  /SVya)  =  0. 

Hence  these  vectors  are  coinplanar,  and  the  planes  therefore 
intersect  in  a  right  line. 

Otherwise:  from  Equation  (111) 

V(aV^y)  =  ySa/?-/?S.-zy; 


162  QUATERNIONS. 

hence,  from  (6'), 

S(pySayS  -  pjSSay)  =  SaySSpy  -  SaySp^  =  0. 

Similarl}-,  or  by  cyclic  permutation, 

SySySpa  -  S/SaSpy  =  0, 
SyaSpyS  -  Sy/?S,oa  =  0. 

But  the  sum  of  these  three  equations  is  identically  zero,  either 
two  giving  tlie  tliird  by  subtraction  or  addition. 

17.  To  Jiiid  the  locus  of  a  point  v:liich  divides  all  right  lines 
terminativg  in  two  given  lines  into  segments  tvhich  have  a  com- 
mon ratio. 

Fig.  67.      ^  Let  DA  and  d'b  (Fig,  67)  be  the  two 

jj   ^__,,--'     I  given  lines,  a  and  (3  unit  vectors  parallel 

to  them,  MX  any  line  terminating  in  the 
^  given    lines,    and    k    a   point    such    that 

^jy- L RX  =  muR.     Assume  do',  a  perpendicular 

to    both   the   given   lines,   o,   its    middle 
point,  as  the  origin,  and  od  =  S,  od'  =  —  S,  or  —  p. 
Then 

OA  =  p  +  KA  =  8  +  Xa. 

OB  =  p  +  UU  =  -  S  +  i/fS. 
Adding 

2p  +  nx  +  nii  =  .va  +  i/,8.  (a) 

But 

RA  +  iJB  =  '-^^-=^  nx  =  ^'^l^  ^-p^8  +  xa), 
m  m  ^ 

which  substituted  in  {a)  gives 

P  —  S  —  .i-a  =  m  (yp  —  p  —  8) ,  {h) 

whence,  since  S5/S  =  S3a  =  0. 

S5p(m  +  1)  =  8-(1  -m)  =  c, 
or  the  locus  is  a  plane  perpend icidar  to  dd'. 


APPLICATIONS    TO    LOCI.  163 

If  the  given  ratio  is  unit}-,  or  r.u  =  ua,  tlion  vi  =  1  and 

S8p  =  0, 

and  the  locus  is  a  |)lane  tin-ougii  o  perpendicular  to  dd'. 
If  a  and  (3  are  parallel,  then  {b)  becomes 

^j  —  S  =  ill  {x'a  —  p  —  8) , 
whence 

)S5,o(m  +  1)  =  (1  -m)?r, 

a  right  line  perpendicular  to  dd'.     If  at  the  same  time  ?u  =  1, 

S3/)  =  0     and     p  —  .r"a, 

a  right  line  through  the  origin  parallel  to  the  given  lines. 

18.  If  the  suvis  of  the  perpendiculars  froyn  two  given  points  on 
tivo  given  jyJnnes  are  equal,  the  sum  of  the  perpendicxdars  from 
any  point  of  the  line  joining  them  is  the  same. 

Let  A  and  u  be  the  given  points,  oa  =  a,  ob  =  /?,  and  S^p  —  d, 
S5'p  =  d'  be  the  equations  of  the  planes  ;  8  and  8'  being  unit 
vectors,  so  that  a;S  and  yS'  are  the  vector-perpendiculars  from  a 
on  the  planes.     Then 

X  =  SaS  -  f/, 

and 

x  +  y  =  S,(?>  +  ?>')-{d-\-d'). 
Similarly 

x'+y'=S/3{8  +  ^')-{d  +  d'). 

But,  by  condition, 

Sa(8  +  S')  =  S/3(8  +  ,^/), 
or 

S(/?-a)(8-f-S')  =  ().  (a) 

The  vector  from  o  to  any  other  i)oint  of  the  line  ah  is 
a  +  2!  (/8  —  a)  ;  whence,  for  this  point, 

.r"  +  y"  =  S[a+z(f3-  a) ]  (8  +  8')  -  {d  +  d') , 

for  which   point,   since  {a)   ivmains  true,  the  sum  therefore  is 
unchanged. 


164  QUATERNIONS. 

19.  To  find  the  locus  of  the  middle  points  of  the  elemods  of  an 
hyperbolic  p)araholokl. 

Let  the  equations  of  the  plane  director  aud  rectilinear  direc- 
trices be 

S5p  =  0, 
p  =  a  -f  .r/S     and     p  =  a'  +  .i-'/S! 

Also,  let  0M  =  /x  be  the  vector  to  the  middle  point  of  an  cle- 
ment so  chosen  that  the  vectors  to  the  extremities  are  a  +  .r/3 
and  a'  -f  x'ji'.     Then,  since  m  is  the  middle  point, 

2  /x  =  a  -I-  a-yS  +  a'  +  x'0.  (a) 

The  vector  element  is 

—  x'P'  —  a'  +  a  +  if/3, 

and,  being  parallel  to  the  plane  director, 

SS(  -  a'  +  a  +  .r/5  -  .t'/S')  =  0. 

This  is  a  scalar  equation  between  known  quantities  from  which 
we  may  find  x'  in  terms  of  x ;  substituting  this  value  in  (a),  we 
have  an  equation  of  the  form 

p,  =  tti  +  .T^i, 

or  the  locus  is  a  right  line. 

20.  If  from  any  three  points  on  the  line  of  intersection  of  tivo 
planes,  lines  be  draion,  one  in  each  plane,  the  triangles  formed 
by  their  intersections  are  sections  of  the  same  pyramid. 

The  Circle  and  Sphere. 

78.    E(piations  of  the  circle. 

The  equation  of  the  circle  may  be  written  under  various 
forms.  If  a  and  fB  are  vector-radii  at  right  angles  to  each  other, 
and  Ttt  =  T/3,  we  ma}'  write 

p  =  cos^  .  a  +  sin^  .  /3    .     .     .     .      (203), 

in  terms  of  a  single  variable  scalar  6. 


APPLICATIONS    TO    LOCI. 


1G5 


If  a  and  /5  arc  unit  vectors  along  the  radii, 

p  =  xa  +  .v/i, 
or,  since  ar  +  ?/-'=  ?", 

p=.(r-/)^«  +  # 
The  initial  point  being  at  the  center. 


(-204). 


(•2().-.) 


=  -r'} 


are  evidently  all  equations  of  the  circle. 

If  o  (Fig.  68)  be  any  initial  point,  c  the  center,  to  which  the 
vector  oc  =  y,  p  the  variable  vector  to 
any  point  p,  cp  =  a,  then 

p-y=a, 

whence 

(p-yy-  =  -^~  -    •  (-^06), 

the  vector  equation  of  the  circle  whose 
radius  is  /•. 

If  Ty  =  c,  it  may  be  put  under  the  form     J 

p''-2Spy  =  c--r- (-207) . 

If  the  initial  point  is  on  the  circumference,  we  still  have 
(p  —  y)-  =  —  7"  ;  but  y'  =  —  '",  hence 

p2_.2S;jy  =  0 (-208), 

or,  since  in  this  case  Spy  =  Spa, 

p-  —  2  Spa 


0 


(20!)). 


79.   Equations  of  (he  sphere. 

This  surface  may  be  conveniently  treated  of  in  connection 
with  the  circle  ;  for,  since  nothing  in  the  previous  article  restricts 
the  lines  to  one  plane,  the  equations  there  deduced  for  the  circle 
are  also  applicable  to  the  sphere. 


QUATERNIONS. 


Another   convenient   form    of  the    equation   of  a   sphere    is 
(Fi^.  68) 

T(p-y)  =  Ta (210), 

the  center  being  at  the  extreniitj-  of  y  and  Ta  the  radius. 

80.    Tangent  line  and  lolane. 

A  vector  along  the  tangent  Ijeing  rf^,  we  have,  from  Equation 
(203), 

dp  —  —  sin^  .  a  +  cos^  .  ^, 

and  for  the  tangent  line  -n-  =  p  +  xdp. 

7r=cos^  .  a  +  sin(9  .  (3  +  x[-sm6  .  a  +  co.s6J  .  /3]    (211), 

where  tt  is  any  vector  to  the  tangent  line  at  the  point  corre- 
sponding to  6. 

From  the  above  we  have  directl}' 

Hpdp  =  0, 

or  the  tangent  is  perpendicular  to  the  radius  vector  drawn  to  the 
point  of  tangency. 

B}-  means  of  this  propert}'  we  ma}' 
-B  write  the  equation  of  the  tangent  as 
follows  :  let  tt  be  the  vector  to  any  point 
of  the  tangent,  as  b  (Fig.  G9),  c  being 
the  initial  point  and  p  the  vector  to  p, 
the  point  of  tangency.     Then 


whence 


S,o(7r- 

p) 

=  0, 

S,07r  = 

— 

1 

TT 

s-  = 
p 

1 

L 

(212), 


are  the  equations  of  a  tangent.  Since  nothing  restricts  the  line 
to  one  plane,  they  are  also  the  equations  of  the  tangent  plane  to 
a  sphere. 


APPLICATIONS   TO   LOCI.  167 

The  above  well-known  property  nia}-  also  be  obtained  I)\- 
dirterentitvting  T/j  =  Ta  ;  whence,  Art.  67,  2, 

Sijdp  =  0, 

and  therefore  p  is  perpendicular  to  the  tangent  line  or  phuie. 

81.  Cliords  of  contact. 

In  Fig.  0!)  let  en  =  /3  be  the  vector  to  a  given  point.  The 
equation  of  the  tangent  bp  must  be  satisfied  for  tliis  i)oint ; 
hence,  from  Equation  212, 

or 

S^o-=-r2 (210), 

which  is  equally  true  of  the  other  point  of  tangencj-  p,'  and  ])eing 
the  equation  of  a  right  line,  it  is  that  of  the  chord  of  contact  ppI 
And  for  the  reason  previousl}'  given,  it  is  also  the  equation  of 
the  plane  of  the  circle  of  contact  of  the  tangent  cone  to  the 
sphere,  the  vertex  of  the  cone  being  at  b. 

82.  Exercises  and  Problems  on  the  Circle  and  the 
Sphere. 

In  the  following  jiroblems  the  various  equations  of  the  plane, 
hue,  circle  and  sphere  are  employed  to  familiarize  the  student 
with  their  use.  Other  equations  than  those  selected  in  any 
special  problem  might  have  been  used,  leading  sometimes  more 
directly  to  the  desired  result.  It  will  be  found  a  useful  exercise 
to  assume  forms  other  than  those  chosen,  as  also  to  transfoinn 
the  equations  themselves  and  interpret  the  results.  Thus,  for 
example,  the  equation  of  the  circle  (209), 

p-  -  2  Spa  =  0 

may  be  transformed  into 

Sp(p-2a)  =  0, 


108 


QUATERNIONS. 


which  gives  immediately  (Fig.  70)  the  property-  of  the  circle, 
that  the  angle  inscribed  in  a  semi-circle  is 
a  right  angle.  Obviousl}',  this  includes  the 
case  of  chords  drawn  fi'om  any  point  in  a 
sphere  to  the  extremities  of  a  diameter,  and 
the  above  equation  is  a  statement  of  the  prop- 
osition that,  p  being  a  variable  vector,  the 
locus  of  the  vertex  of  a  right  angle,  whose 
sides  pass  through  the  extremities  of  a  and 
^  —  a,  is  a  sphere. 

Again,  with  the  origin  at  the  center,  we  have  (Fig.  71), 

(^.  +a)+(a-p)=2a, 

and,  operating  with  x  S  .  (p  —  a), 

S(p  +  a)(p-a)  =  0; 

.-.  p  is  a  right  angle.     This  also  follows  from 
Tp—Ta,  whence  p-=a^  and  S(p+a)(p— a)  =  0. 
Again,  from  Tp  =  Ta, 


T(p  +  a)(p-a): 


:TV. 


ap. 


The  first  member  is  the  rectangle  of  the  chords  pd,  pd'  (Fig.  71), 
and  the  second  member  is 


2  CD  .OP  sinDOP. 

Hence  the  rectangle  on  the  chords  drawn  from  any  point  of  a 
circle  to  the  extremities  of  a  diameter  is  four  times  the  area  of 
a  triangle  whose  sides  are  p  and  a. 

Also,  from  Tp  =  Ta, 

and  for  any  other  point 

p'2  =  -7-; 

•••  Hp'-\-p)(p'-p)  =  o. 

But  p'—  p  is  a  vector  along  the  secant,  and  p'+  p  is  a  vector 
along  the  angle-bisector ;  now  when  the  secant  becomes  a  tan- 


ArPLICATIONS    TO    LOCI. 


169 


gent,  the  angle-bisector  becomes  tlie  nulius  ;  therefore  the  rachiis 
to  the  point  of  contact  is  perpcndicnhir  to  the  tiingent. 

1 .    Tlie  anyle  at.  the  center  of  a  circle  is  double  that  at  the  cir- 
cumference stcoidiiip  on.  the  same  arc. 


Tp  =  Ta, 


We  have 
and  therefore,  Art.  oG,  18, 

p  =  (p4-tt)"'a(p  +«), 

whence  the  proposition. 

2.  In  any  circle,  the  square  of  the  tangent  equals  the  product 
of  the  secant  and  its  external  segment. 

Fig.  C9  (bis). 

In  Fig.  69  we  have  p 

CB  =  CP  -f-PB, 
.'.    CB'-=  CP-+ Pli'^, 

or 

PB-=  cb'—  cp- 

=  CB-  — cir,  as  lines, 
=  BD  .  bd'. 

3.  The  right  line  joining  the  points  of  intersection  of  two  circles 
is  perpendicular  to  the  line  Joining  their  centers. 

Let  (Fig.  72)  cc'  =  a,  cp  =  p,  cp'  =  p',  and  r,  r'  be  the  radii 
of  the  circles.     Then 


r  =  -r. 

(p-aY  =  -r"-; 

also 

{p'-ar-  =  -r'\ 

Hence 

Spa  =  S,o'a, 

or 

Sa(p-p')  =  0; 

hence  pp 

and 

cc'  are  at  right  angles 

170  QUATERNIONS. 

4.  A  chord  is  drawn  imrallel  to  the  diameter  of  a  circle;  the 
radii  to  the  extremities  of  the  chord  make  eqncd  angles  with  the 
diameter. 

If  p  and  p  be  the  vector-radii,  2  a  the  vector-diaineter,  then 
ica  =  the  vector-cliord,  and 

{p—  xa)-  =  —  ?-^, 
(p+.ra)-  =  -?-2, 

whence  the  proposition. 

5.  //' ABC  is  a  triangle  inscribed  in  a  circle,  shoiv  that  the  vector 
of  the  product  of  the  three  sides  in  order  is  loarallel  to  the  tangent 
at  the  initial  2^oint.      [Compare  Art.  55.] 

If  AB  =  /3,  CA  =  y,  and  o  is  the  center  of  the  circle,  then 

- V(AB   .  BC   .  CA)  =  V  .  ;8(/3  +  y)y 
=  V(^^'y+/3/) 

=  fi-'y  +  r/3. 

Or,  c  and  b  being  points  of  the  circumference  satisfying 
p^  — 2S/)a  =  0  [Eq.  (209)],  substituting  and  operating  with 
S  .  aX 

S  .  aV(AB  .  BC  .  Ca)  =  2  Sa^Say  -  2  Sa/3Say  =  0. 

Hence  V(ab  .  bc  .  ca)  is  perpendicular  to  a,  or  parallel  to  the 
tangent  at  a. 

6.  The  Slim  of  the  squares  of  the  lines  from  any  jwint  on  a 
diameter  of  a  circle  to  the  extremities  of  a  jmrallel  chord  is  equal 
to  the  sxwi  of  the  squares  of  the  segments  of  the  diameter. 

Let  pp'  (Fig.  73)  be  the  chord  parallel  to  the  diameter  ddJ 
FifT.  73.  o  the  given  point,  and  c  the  center  of  the 

circle.     Let  cp  =  p,  cp'  =  p',  oc  =  a,  op  =  )8 
and  op'  =  /?!     Then 

Op2  =  -^^  =  -(a2  +  2Sap+p2)^ 

OP'2  =  _^'^'=_(a2+2Sap'+p'2); 

.-.  0P2  +  OP'-  =  2oc-  +  2  DC-  -  2 (Sap  +  Sap'). 


APPLICATIONS   TO    LOCI.  171 

But 

Therefore 

Sap  +  Sa/a'  ^=  0, 
.111(1 

OP-  +  op'-  =  DO-  +  Ol)'-. 

7.  To  find  the  intersection  of  a  plane  and  a  sphere. 

Let  p-  =  —  1^  be  the  equation  of  the  sphere,  8  a  vector-perpen- 
(licuUir  from  its  center  on  the  plane  and  T8  =  (/.  Then,  if  ji  \k\ 
ii  vector  of  the  plane, 

P  =  8  +  ^. 

Substituting  in  the  equation  of  the  sphere,  since  S/3S  =  0,  we 
have 

the  equation  of  a  circle  whose  radius  is  V?-  —  d'\  and  which  is 
real  so  long  as  d  <  r. 

8.  To  find  the  intersection  oftioo  spheres. 

Let  the  equations  of  the  given  spheres  be  (Eq.  207) 

p-  -  2  Spy  =  c-  -  ?-2, 
p'--2Hpy'=c"-r'K 

Subtracting,  we  have 

2  Spiy  —  y')  =  ct,  constant. 

The  intersection  is  therefore  a  circle  whose  plane  is  perpen- 
dicular to  y—  y'  the  vector-line  joining  the  centers  of  the  spheres. 
Assuming  (Eq.  210) 

T(p-y)  =  Ta     and     T(p-y')  =  Ta,' 

show  that  2S/j(y  —  y')  =  a  constant.,  as  above. 


172  QUATERNIONS. 

9.  The  planes  of  intersection  of  three  spheres  intersect  in  a 
light  line. 

Let  y',  y",  y'"  he  the  vector-lines  to  the  centers  of  the  spheres, 
and  their  equations 

fj-  —  2  Spy'  =  c', 
p--2Spy"  =  c", 

p2_2Spy"'=c'". 

The  equations  of  the  planes  of  intersection  are,  from  the  pre- 
ceding problem, 

2Sp(y'-y")=c"-c',  (a) 

2Sp(y'-y"')  =c"'-c',  {b) 

.        2Sp(y"-y"')=c"'-c".  (c) 

Now,  if  p  be  so  taken  as  to  satisf}'  (a)  and  (6),  we  shall 
obtain  their  line  of  intersection.  But  if  p  satisfies  (a)  and  (b), 
it  will  also  satisfy  their  difference,  which  is  (c) ;  the  planes  there- 
fore intersect  in  a  right  line. 

10.  To  find  the  locus  of  the  intersections  of  perp>endiculars  from 
Q,  fixed  p)oint  iqjoji  lines  through  another  fixed  point. 

Let  p  and  p'  be  the  points,  pp'  =  a,  and  8  a  vector-perpen- 
dicular on  an}'  line  through  pj  as  p  =  a  -f  x[i.     Then 

8  =  a  +  7//3, 
and  operating  with  S  .  8  x 

h-  =  SSa, 

which  is  the  equation  of  a  circle  (Eq.  209)  whose  diameter  is  pv. 

11.  From    a  fixed  j'^oint  p,   lines   are   drawn  to  points,   as 

p',  p",  of  a  given  right  line.     Required  the  locus  of  a  point  o 

071  these  lines,  such  that  pp'  .  po  =  ni-. 

Let  the  variable  vector  po=p  ;  then  pp' =a;p.    By  the  condition 

T(pp' .  Po)=     m', 
or 

T(xp.p)=     m'; 
.•.  xp-  =  —  nr. 


APrLICATIONS   TO   LOCI.  173 

If  8   bo   the  vcctor-porijcndicular   IVom    v  on   the  given   line, 

and  T8  =  d, 

S5(.rp-8)  =  0, 

or 

xSBp  =  —  (J- ; 

hence  the  locus  is  a  circle  through  r. 

12.  If  through  any  point  chords  be  dratvn  to  a  circle,  to  find 
the  locus  of  the  intersection  of  the  pairs  of  tangents  drawn  at  the 
points  of  section  of  the  chords  and  circle. 

Let  the  point  a  be  given  by  the  vector  oa  =  a,  o  being  the 
initial  point  taken  at  the  center  of  the  circle.  Let  p'  =  ok  be 
the  vector  to  one  point  of  intersection  R.  The  locus  of  r  is 
required.     The  equation  of  the  chord  of  contact  is  (Eq.  21:3) 

Sp'o-  =  -  r, 

which,  since  the  chord  passes  through  a,  may  be  written 

Hp'a  =  -r\ 

where  a  is  a  constant  vector.     The  locus  is  therefore  a  straight 
line  perpendicular  to  oa  (Eq.  IHQ). 

13.  To  find  the  locus  of  the  feet  of j)eri')endicidars  draton  through 
a  given  point  to  planes  passing  through  another  given  point. 

Let  the  initial  point  be  taken  at  tlie  origin  of  perpendiculars, 
a  the  vector  to  the  point  through  which  the  planes  are  passed, 
and  8  a  perpendicular.     Then 

S3(8-a)=0, 

or 

S-'_Sa8=0 

is  true  for  any  perpendicular.    Hence  the  locus  is  a  sphere  whose 
diameter  is  the  line  joining  the  given  points. 


17-J:  QUATERNIONS. 

Otherwise  :  if  the  origin  be  taken  at  the  point  common  to  the 
planes,  and  the  equation  of  one  of  the  planes  is  S5p  =  0,  then 
the  vector-perpendicular  is  (Eq.  198) 

and,  if  p  be  the  vector  to  its  foot, 

p  =  a-8-iSSa, 

or 

p  —  a  =      —  S"^SSa, 

whence 

{p-a)-=       8--(SSa)2, 
and 

Sap  —  a-  =  —  8~-(S8a)^. 

Adding  the  last  two  equations 

p' —  Sap  =  0, 

or 

T(p-ia)  =  Tia, 

which  is  the  equation  of  a  sphere  whose  radius  is  T-  and  center 
is  at  the  extremity  of  ^,  or  whose  diameter  is  the  "hue  joining 
the  points. 

14.  To  find  the  locus  of  a  2'>oint  p  ivJiich  divides  any  line  os 
draivnfrom  a  given  point  to  a  given  plane,  so  that 

OP  .  OS  =  ?>i,  a  constant. 

Let  OP  =  p  and  os  =  o- ;  also  let  SScr  =  c  be  the  equation  of  the 
plane.     We  have,  bv  condition, 


and 


and 


TpTo- 

=  m, 

Up 

=  U(7; 

'.   To- 

m 
~Tp' 

(T  ■■ 

mVp 
~    Tp 

mp 
P'' 

APPLICATIONS   TO    LOCI.  175 

Siil)stitiitin_<2;  in  the  c'(|ti;iti<)n  of  the  plane 

viSSfj  +  ''/^'  =  0, 

which  is  the  equation  of  a  spliere  passing  through  o  and  having 
—  for  a  diameter. 

OD 

15.    To  find  the  locus  of  a  point  the  ratio  of  irhose  distances 
from  two  given  j)oints  is  constant. 

Let  o  and  a  be  the  two  given  points,  oa  =  n,  ok  =  p,  r  being 
a  point  of  the  locus.     Then,  b^-  condition,  if  m  be  the  given 

ratio, 

T(p  — a)  =  mTp, 
whence 

p'  —  2  Spu  +  a'  =  nc  p", 

(1  —  m-)p'  =  2  Sap  — 


1  -  m- 

=  z  ^ap  — 

2  Sap 


'  1  -  1)V 


+ 


l-m^      (1-w-)-      (1-m-)- 


T    P-T-^,HT 


ma 


l—m'-J  1  —  m- 

which  is  the  equation  of  a  sphere  whose  radius  is  T ;a,  and 

-.        1  —  m- 

whose  center  c  is  on  the  line  oa.  so  that  oc  = -a.    (Kq.  210) . 

l  —  iu- 

16.    Given  two  points  a  and  p.,  to  find  the  locus  of  v  irhen 

Av"-  +  r.p-  =  OP-. 

o  being  the  origin,   let  oa  =  a,   or.  = /3,   op=p.     Then,  by 
condition, 

p-  =  (p-a)-  +  (p-^)% 

whence 

p2_2Sp(a  +  /3)  =  -(a-  +  /5--'), 
[p-(a  +  ^)]'^=2Sa^, 
T[p-(a  +  /S)]=V-2Sa^, 


176  QUATERNIONS. 

which  is  the  equation  of  a  spliere  wiiose  center  is  at  the  exti-eniity 
of  (a  +  /S),  if  Sa/3  is  negative,  or  the  angle  aob  acnte.  If  this 
angle  is  obtuse,  there  is  no  point  satisfying  the  condition.  If 
AOB  =  90°,  the  locus  is  a  point. 

83.   Exercises    in   the   transformation   and    interpretation    of 
elementary  symbolic  forms. 

1.  From  the  equation 

derive  in  succession  tlie  equations 

T(p  +  a)  =  T(p-a)     and     T^^=  1, 
^r         '         \i  p  _  t^         . 

and  state  what  locus  they  represent. 

2.  From  the  equation 

a       a 

derive  symbolically  the  equations 
ap  +  pa  =  0,   S-  =  0,  SU-  =  0,  (u-Y==-l,  and  TyU-  =  l, 


and  interpret  them  as  the  equations  of  the  same  locus. 
3.    Transform 


to  the  forms 


p  —  a 

s'^ =  0 


S-=:l     and     SU-=T 


and  intei'prct.  ""  -I 

4.    Transform   s'^-^-^^O  to  S-=S-'  -'^"c^  interpret 


5.    Transform   (p -/S)"  =  (p  -  a)-   to    T  (p -/3)  =  T  (p  -  a). 
and  interpret. 

G.    What  locus  is  represented  by  K =  0? 

^  -^        a       a 


APPLICATIONS    TO    LOCI.  IT' 


8.  AVhatby  U^=:r^?     Vp^VfS't     uf=l? 

a  a  p 

P  (3. 

9.  U-  =  -U-? 


10.  (un'^u^? 


11.  v'-^^  =  o?   y^  =  v^? 


12.    V-=0? 


13.    ^K^  =  rr? 


^TTe_«lT??       «TTe__^iTT^V       fKv'i 


14.  su-  =  su-?    su-  =  -sir-?     su'-   =  siJ 

a  tt  a  a         \        aj 

15.  Tp  =  l? 

16.  Transform  (p  —  a)-  =  a-  to  T(p  —  a)  =  Ta,  luid  interpret. 

17.  Under  what  other  form  may  we  write  {p  —  u)-=  (/5—  <i-)-'^ 
Of  what  locus  is  it  the  equation  ? 

18.  p-  +  a=  =  0?  /3-+l=0?  Translate  the  hitter  into  Car- 
tesian coordinates,  b}"  means  of  the  trinomial  form,  and  so  deter- 
mine the  locus  anew. 

19.  T(p-/3)  =  T(/3-a)? 

20.  Compare  SU-  =  T-  and  S-  =  1  with  the  forms  of  Ex.  3. 

pa.  p 

21 .  Wliat  locus  is  represented  by  S/8p  -f-  p'  —  0  when  T/3  =  1  ? 

2..  r.-Y-rv'-Y=.v 


24.    Show   tluit  V  .  Va/3Vap  =  0  is  the  equation  of  a   i)lanc. 
What  plane?     [Eq.  (112)]. 


178  QUATERNIONS. 

The  Conic  Sections. 
Cartesian  Forms. 
84.    The  Parabola. 
Resuming  the  general  form  of  the  equation  of  a  plane  curve 

from  the  relation  y-  =  2px,  we  obtain 

for  the  vector  equation  of  the  parabola  when  the  vertex  is  the 
initial  point.  If  the  latter  is  taken  anywhere  on  the  curve,  from 
the  relation  y-  =  '2p'.v,  we  obtain 

P  =  f^,«  +  Z//8 (-210); 

and  if  the  initial  point  is  at  tlie  focus,  then  //- =  2px  +/r  gives 

P  =  ^{lf-ir)o^  +  l/(3      ....    (216); 
2/> 

or  again,  in  terms  of  a  single  scalar  t, 

P  =  -,a  +  t(^ (217). 

In  Equations  (214),  (21.5)  and  (216),  a  and  (3  are  unit  vectors 
parallel  to  a  diameter  and  tangent  at  its  vertex,  being  at  right 
angles  to  each  other  in  P2quations  (214)  and  (216) ;  in  Equation 
(217)  a  and  (3  are  any  given  vectors  parallel  to  a  diameter  and 
tangent  at  its  vertex,  the  initial  point  lieing  on  the  curve. 

85.    Tangent  to  the  parabola. 

From  Equation  (216)  we  have  for  the  vector  along  the  tangent 
(Art.  62) 


APPLICATIONS   TO    LOCI.  1T9 

and,  therefore,  the  equation  of  the  tangent  is 

^  =  i-(2/^-p-^)a  +  ///i+r(^^.  +  /i)      .      .    (218). 
From  Equation  (217)  the  vector  along  the  tangent  is 
ta  +  13, 
and  the  equation  of  the  tangent  is 

^  =  fla  +  t(3  +  Afa+(^)      ....     (-.^10). 

If  p  be  the  A-eetor  to  a  point  on  the  diameter  of  a  parabola,  the 
point  being  given  by  the  equation 

p=ma  +  nf3,  (o) 

and  a  tangent  to  the  curve  be  drawn  through  this  point,  then 
{((}  must  satisfy  the  equation  of  the  tangent-line  and 

ma  +  U(3  =  ^-a  +tft  +  X{ta  +  (3)  , 

whence  ^2 

m  =  -  +  .ft     and     ji  =  t  -^  .r, 

^^  t  =  n±  V»-'-2m ;  {b) 

hence,  in  o-enei'al,  two  tangents  can  be  drawn  to  the  ciu've  through 
the  given  point.  AVhen  }r  =  2m,  they  coincide  ;  in  this  case, 
from  (a),  ^^ 

P  =  -«  +  "/?. 

tlie  point  being  on  the  cun'e.  If  2m>n-,  t  is  imaginary,  and 
no  tangent  can  be  drawn  ;  in  this  case  (a)  becomes 


P=l-+a)a  +  nl3, 
the  point  being  within  the  curve. 


180 


QUATERNIONS. 


86.   Examples  on  the  parabola. 

1.    The  intercept  of  the  tangent  on  the  diameter  is  equal  to  the 


Fig.  74. 


abscissa  of  the  point  of  contact. 

Since  the  tangent  is  parallel  to 
the  vector  ta  +  /8,  or  to  an}-  multiple 
of  it,  it  is  parallel  to  t-a-\-t/3  or  to 

-a-^tf3  +  ~a,  that  is,  to  (Fig.  74) 


But 


OP  4-  ox. 


TP  =  TO  +  OP  ; 
TO  =  ox. 


2.    If,  from  any  point  on  a  di- 
ameter jjrodiiced,  tangents  be  draivn, 
the  chord  of  contact  is  parallel  to  the 
tangent  at  the  vertex  of  the  diameter. 
If    t'   and   t"  correspond    to   the 
points  of  tangency,  we  have  for  the  vector-chord  of  contact 


P  -  P 
which  is  parallel  to 


2  ^        2 


t'+t' 


t"(B, 


or,  from  Equation  (&),  Art.  8.'>,  to 

which  is  independent  of  m. 

3.    To  find  the  locus  of  the  extremity  of  the  diagonal  of  a  rect- 
angle whose  sides  are  two  chords  draivn  from  the  vertex. 

Let  OP  and  op'  he  the  chords.     Then 


OP  =  p 


2p 


+  llf^^ 


OP'=p'=:— a-  ?/'/?. 
'>p 


(«) 
(6) 


APPLICATIONS    TO    LOCI.  181 

The  vectoi'-diagoiuil  w'  is  p  +  p',  or 

which  may  be  put  uiuUt  the  form  of  the  equation  of  the  parabola 

'^  ini'         ■    . 
by  adding  and  sul)ti-actiMg  ^  u,  givmg 

But,  by  condition,  Spp' =  0.  Hence,  from  (a)  and  (ft),  Sa^ 
being  zero, 

which  in  (c)  gives 

^'  =  ^^^^^'a  +  (?y  -  2/')i8  +  ^pa. 

Changing  the  origin  to  the  extremity  of  4/)a, 

Hence  the  locus  is  a  similar  parabola  whose  vertex  is  at  a 
distance  of  twice  the  parameter  of  the  given  parabola  from  its 
vertex. 

Moreover,  from  {d),  .rx' =  (2;))-.  Hence  the  parameter  is  a 
mean  proportional  beticeen  the  ordinates  and  the  abscissas  of  the 
extremities  of  chords  at  right  angles. 

4.  If  tangents  be  draiim  at  the  vertices  of  an  inscribed  triangle, 
the  sides  of  the  triangle  produced  will  intersect  the  tangents  in  three 
points  of  a  right  line. 

Let  opp'  (Fig.  74)  be  the  inscrilied  triangle,  and  one  of  the 
vertices,  as  o,  the  initial  point.  Then,  for  the  points  p  and  p' 
respectively,  we  have 

P=^\  +  t(i, 


182  QUATERNIONS. 

Let  TTi,  TTj,  TTg  be  the  vectors  to  the  points  of  intersection  ;  tlien 

xi  =  OP  +  PS,  =  ^"a  + //3  +  .r(?a  + /3)  . 

Also 

TTi  =  a-'op'=  .^''(  — a  +  ^'/3 


t-  r'f'- 

L^xt  =  '^,     t  +  x  =  x'tl 


X   = 


•>tt'-t'- 

Hence 


-5^(f+'''')=^{r+^} 


In  a  similar  manner 


'"  ^^«+, 


But 


Also 


Hence 


Now 


TTs  =  OP  4-  y^y'  =  OP  +  y{p—p) 
IT,,  =  z(i ; 
—  _^ 


'■3  =  7— T-,p- 


«  +  r 


■^„-i^'.,_?!llL%=,(^'„+/3)-c(^a+/j)-(/-<')/3=0, 

Also 

2^-^'  _  2^'-<  _  t--  t'-  ^  Q 
^  t'  W 

Hence  ttj,  TTg  and  -n-.^  terminate  in  a  straight  line. 


APPLICATION!^    TO    LOCI. 


183 


o.    The  jvincipaJ  tavf/pnf  As'  trDir/ctit  to  all  circles  described  on 
the  radii  veetores  as  diameters. 

Let  Ai'  =  p  (Fig.  7o),  a  and  /3 
being  unit  vectors  along  tlie  axis 
and  principal  tangent.  Then,  if 
the  circle  cnt  tlie  tangent  in  r, 
and  TC  be  drawn  to  the  center. 

T(TC)  =  T(KC)  =  T(^KP)  ; 

.-.    TC- =  j(p  —  7Ha)-. 

Also 

TC  =  TA  +  AF  +  FC 

=  —  z(3  +  ma  +  i(p—  ma). 

TC-  =  \_—zl3-\-  Vla-{-^{p  —  ma)']-. 

pjQuating  these  values  of  tc", 
we  have,  since  S/3a  =  0, 

z-l3--zSf3p  +  mHap  =  0, 


z--z>j-\- 


0, 


wliich  o;ives  but  one  value  for  z. 


6.    To  find  the  length  of  the  curoe. 

It  has  been  seen  (Art.  G2)  that,  if  p  =  c^(/)  ho  the  equation 
of  a  plane  curve,  the  differential  coefficient  is  the  tangent  to  the 
curve.  Hence,  if  this  be  denoted  b}'  p'  =  ({,'(t),  Tp'dt  is  an 
element  of  the  curve  whose  length  will  l)e  found  b\-  integrating 
T/j'  with  reference  to  the  scalar  variable  involved  between  proper 
limits  :  or 


.,,=j;'t,/. 


For  the  parabola 


a +  1,13, 


184  QUATERNIONS, 

we  have 

,   y    .  ^ 
p 


p 


P'-'lh  L  -P  -  \  J'  yji/n 

7.     To  find  the  area  of  the  curve. 

With  the  notation  of  the  previous  example,  twice  the  area 
swept  over  by  the  radius  vector  will  be  measured  by  (Art.  41,7) 
TYfjp'cU.  The  area  will  then  be  found  by  integrating  TYpp'  with 
reference  to  the  involved  scalar  between  proper  limits  and  taking 
one-half  the  result ;  or  * 

A-A,  =  ^Jt\pp'. 

For  the  parabola 

.-..„=ijr;^v(i;„..,)(;w4 

or,  since  a/S  =  1)0°, 


From  the  origin,  where  ?/„  =  0,  to  any  point  whose  ordinate  is 
?/,  the  area  of  the  sector  swept  over  b}-  p  is f'  =  ^xij  ;  adding 

the  area  ^.vy  of  the  triangle,  which,  with  the  sector,  m^kes  up 
the  total  area  of  the  half  curve,  we  have  ^xy,  or  two-thirds  that 
of  the  circumscribing  rectangle.  The  origin  may  be  changed  to 
an}-  point  in  the  plane  of  the  curve,  to  which  the  vector  is  y,  by 
substituting  the  value  p  =  y  +  pi  in  the  equation  of  the  curve, 
Pi  being  the  new  radius  vector  ;  we  may  thus  find  any  sector  area 
limited  bj'  two  positions  of  pi,  the  vertex  of  the  sector  being  at 
the  new  origin.  Thus,  transferring  to  an  origin  on  the  principal 
tangent,  distant  b  from  the  vertex,  p  =  6/5  +  pi;  which,  in  the 
equation  of  the  parabola,  gives 


APPLICATIONS    TO   LOCI. 


185 


integrating,  as  before,  between  the  limits  ij  =  b  and  tj  =  0, 

1 


(\p 


//■'  =  1T-1V/- 


87.    Relations  betit-een  three  intersecttwj  tancjents  to  the  Parabo- 
la.    [^"Am.  Journal  of 
Math.,"  vol.  i.  p.  379. 
M.      L.     Ilohnan     and 
E.  A.  Englev.] 

Let  pi,  P21  Pi  be  the 
vectors  to  the  three 
points  of  tangency,  Pi, 
Po,  P:(  [Fig.  7G],  and 
TTi,  -21  TTg  the  vectors  to 
Si:  ^25  S3,  tlie  points  of 
intersection  of  the  tan 
gents.  Resuming  Equa- 
tion (216),  wiiere  tlie 
focus  is  the  initial 
point,  and  a  and  (3  are 
unit  vectors  along  the 
axis  and  the  directrix, 


•2p 


(«). 


Since  p^  =  —  (Tp)-,  and  So./?  =  0,  we  liave  for  the  three  points 


Tpi  =  — O/r  +  ^r) 
•2p 

Tp,  =  -^(j/|  +  ir) 
•2p 


(6), 


'rp,  =  —  {>/i  +  F 

2p 
The  vector  along  the  tangent  is 


If 

p      ^A^' 


QUATERNIONS. 


186 

and  therefore 

^i=Ps  +  P3S1  =  ^(2//  -p-)a  +  y-ilS  +  u/^a  +  fs\  ; 
whence,  equating  the  coefficients  of  a  and  jS, 


whence,  substituting,  and  by  the  cyclic  permutation  of  the  sub- 
scrii^ts. 


From  (&) 


and  from  (c) 


'^p-^'^p^^=X^^y"+f){yl+v')  \ 


and  from  (d)  and  (e) 


(T7r3)=^  =  Tp,Tpo 
(T7r2)^=Tp,Tp3 
(Trr,)='  =  T/3,Tp3j 


(0). 


(^0, 


(e), 


(/). 


APPLICATIONS    TO    LOCI. 


187 


From  {(•) ,  it  appears  that  the  distance  of  the  p'tiut  of  intersec- 
tion of  two  tangents  from  the  axis  is  the  arillimelical  mean  of  the 
ordinates  to  their  points  of  contact.  From  (/),  tliat  the  distance 
from,  the  focus  to  the  point  of  intersection  of  tiro  tangents  is  a 
mectn  proportiomd  to  the  radii  vectores  to  the  jxiints  of  contact. 


1st.    If  po  becomes  a  multiple  of  ^, 

p2  =  —{y'i-p')^-\-y-il 
•ip 

.-.    Z  =  ]}.,=  ±p. 


^/3; 


Or,  the  parameter  is  the  double  ordinate  through  the  focus,  or 
tivice  the  distance  from  the  focus  to  the  directrix. 


2(1.    If  pi  is  the  multiple  of  po  (Fig-  '")?  tlieu  p.,  —  p,  is  a  focal 
chord,  and 

.r/j.  =  p,, 
or,  from  (0)1 

X  — - 0/i  -  p-)  tt  +  .'/2 /?  =  -—  {ill  - p-) a  +  III fS  ; 

L-/'  J    -p 


whence 


.Vi-P-  ^Ih 

yi-p-    y-i 


188  QUATERNIONS. 

or 

2/1  {yi  2/2  +  ir)  =  2/2  (2/1 2/2  +/) » 
and 

2/12/2 +P'=0.  (gr) 

From  (o)  and  (c) 

^^3pi  =  -  :^  (2/22/1  -  p-) -z-  iui  -  p-)  -  i (2/1  -f-  2/2)2/1 

=  -- ^-,(2//  +  i>')  (2/12/2  +  F)  =  0  ;  (/O 

or,  a  line  from  the  foats  to  the  intersection  of  the  tangents  at  the 
extremities  of  a  focal  chord  is  perpendicular  to  the  focal  chord. 
The  vectors  along  the  tangents  are 

Pi  —  T^s     and     p.,  —  TTs, 
and,  from  (/i), 

S(P]  —  TTs)  (p2  —  71-3)  =  Sp,  p2  +  TTg-  =  0, 

or,  the  tangents  at  the  extremities  of  the  focal  chord  are  petpen- 
dicidar  to  each  other. 
Since,  from  (g), 

2/12/2  =  -iA 
we  have 

^3  =  Y]  ^2/1 2/2  -  p')  a  +  i  (2/1  +  2/2)/? 

or,  i/ie  tangents  at  the  extremities  of  a  focal  chord  intersect  on 
the  directrix. 

3d.    If  p2  becomes  a  multiple  of  a  (Fig.  78),  ?/2=  0,  and  from 
(c) 

7^3  =  ^/2/22/i  -  F)«  +  i(2/i  +  2/2)^ 

or,  ^/<e  subtangent  is  bisected  at  the  vertex. 


APPLICATIONS    TO    LOCI. 


189 


Also 


.3_p,  =  -^a  +  |/3-(^^a  +  ^,^ 


=  -57,"- 2^- 
Operating  with  S  .  773  X 

4  4 

or,  a  p<'rj7e?i(?/c?<?ar  /ro7>i  the  focus  on  the  tangent  intersects  it  on 
the  tangent  at  the  vertex. 


Again,  since  ttj  is  parallel  to  the  normal  at  Pi,  the  latter  nuiy 
be  written,  from  (i), 

whence 

z  =  -xL,     V,  =  x-L^, 


190  QUATERNIONS. 

hence,  the  subnormal  is  constant ;   and  the  normal  is  tivice  the 
perpendicular  on  the  tangent  from  the  focus. 
The  normal  at  Pi  uia^-  be  written 

a-TTs  =  —  za  +  pi, 
or 

whence,  from  (/_>), 

a;  =  2,     and     z=  —  {y^-  +  p-)  =  To  ; 

or,  the  distance  fro7n  the  foot  of  the  normal  to  the  focus  equals 
the  radius  vector  to  the  jyoint  of  contact,  or  the  distance  from  the 
point  of  contact  to  the  directrix,  or  the  distance  from  the  focus  to 
the  foot  of  the  tangent. 

The  portion  of  the  tangent  from  its  foot  to  tlie  point  of  con- 
tact may  be  written  za  +  p^,  in  Avhich  z  has  just  been  found. 
Hence 

Za  +  pi  =.  — (?//  +  i;-)a  +  J-^{^J■'  -  /)a  -f-  y,(S, 

or 

the  portion  of  the  tangent  from  the  foot  of  the  focal  perpendicu- 
lar to  the  point  of  contact  is 

or 

or,  comparing  (j)  and  (k),  the  tangent  is  bisected  by  the  focal 
perpendictdar,  and  hence  the  angles  between  the  tangent  and  the 
axis  and  the  tangent  and  the  radius  vector  are  eqtiaf  and  the 
tangent  bisects  the  angle  between  the  diameter  and  radius  vector 
to  the  point  of  contact. 


APPLICATIONS   TO   LOCI.  191 

(/,)  is  also  the  perpendicular  from  the  focus  on  the  normal, 
and  shows  that  tlie  locus  of  the  foot  of  the  j^erpendicuhir  from  the 
focus  on  the  normal  is  a  jfarabola,  tvhose  vertex  is  at  the  focus  of 
the  given  parabola  and  whose  jmrameter  is  one-fourth  that  of  the 
given  piarahola. 

88.   The  Ellipse. 

1 .  Substituting  in  the  general  equation  p  —  xa  -\-  yl3  the  value 
of  1/  from  the  equation  of  the  ellipse  referred  to  center  and  axes 

a-y-  +  b-x-  =  a'-b', 
we  have 

p=xa  +  mh{a--x-)'^/3    ....     (220), 

in  which  m  =  —  and  a  and  /3  are  unit  vectors  along  the  axes. 

For  unit  vectors  along  conjugate  diameters,  the  equation  of  the 
ellipse  becomes 

p  =  xa  +  m"i{a'^-x-)h(3   ....     (221). 

Again,  if  c^  be  the  eccentric  angle,  the  equation  of  the  ellipse 
may  be  written  in  terms  of  a  single  scalar  variable, 

^         /3  =  cosc^  .  aH- sin(^  .  y8    ....     (222). 

2.  From  Eq.  (220)  we  have,  for  the  vector  along  the  tangent, 

a  —  mi{a-  —  X')-iX(3  =  a —  J3  =  a (3 

yvi  V«"  —  X'  y 

=  X  {ya  -  mx(3)  ; 
hence,  for  the  equation  of  the  tangent  line, 

77  =  Xa  +  yfS  +  X  (ya  -  mx/3)     .     .     .     (22.".)  : 
or,  more  simply,  from  Eq.  (222),  the  vector-tangent  is 
—  sin  cf>  .  a  -\-  cos  (fi  ,  /3, 


192  QUATERNIONS, 

and  the  equation  of  the  tangent  is 

TT  =  cos  (/)  .  a  +  sin  (/)  .  /5  +  x(  —  sin  (^  .  a  +  cos  eft  .  /3),  (224). 

Since  —  sin  <p  .  a -\-  cos  ^  .  /3  is  along  tlie  tangent,  cos  ^  .  a  + 
sin^  .  (3  and  —  sin^  .  a  +  cos^  .  /3  are  vectoi's  along  conjugate 
diameters. 


89.   Examples  on  the  Ellipse. 

1.  The  area  of  the  parallelogram  formed  by  tangents  drawn 
through  the  vertices  of  any  pair  of  conjugate  diameters  is  constant. 

We  have  directly 

TV[2(cos<^  .  a  +  sin<^  .  f3)  2(-sin<^  .  a  +  cosc;^  .  /3)] 
=  4TVa^  =  a  constant ; 

namely,  the  rectangle  on  the  axes. 

2.  The  sum  of  the  squares  of  conjugate  diameters  is  constant, 
and  equcd  to  the  suvi  of  the  squares  on  the  axes. 

For,  since  Sa/3  —  0, 

(cos<^  .  a  4-  sin^  .  /Sy-  +(  — sin(^  .  a  +  cos^  .  (3)-  =  a-  +  (Sr . 

3.  The  eccentric  angles  of  the  vertices  of  conjugate  diameters 
differ  by  90? 

The  vector  tangent  at  the  extremit}-  of 

p  =  cos<^  .  a  +  sin<^  . /5  (a) 

is 

—  sin  cji  .  a  +  cos  (f)  .  /S. 

This  is  also  a  vector  along  the  diameter  conjugate  to  p,  and  is 
seen  to  be  the  value  of  p  when  in  (a)  we  vrrite  «/>  +  90°  for  c^. 

4.  The  ecceyitric  angle  of  the  extremity  of  equal  conjugate  diam- 
eters is  45°  and  the  diameters  fall  xipon  the  diagonals  of  the 
rectangle  on  the  axes. 


APPLICATIONS   TO    LOCI,  11):', 

5.  The  line  joinimj  the  points  of  contact  of  taiujoMs  at  riijlit 
angles  to  each  other  is  parallel  to  lite  line  joiniiuj  the  cxtremilics 
of  parallel  diameters. 

6.  Tangents  at  right  angles  to  each  other  intersect  in  the  cir- 
cumference of  a  circle. 

7.  If  an  ordinate  pd  to  the  major  axis  he  produced  to  meet  the 
circumscribed  circle  in  q,  then 

QD  :  I'D  :  :  a  :  h. 

8.  If  an  ordinate  pu  to  the  minor  axis  meets  the  inscribed  circle 

in  Q,  then 

Qi>  :  PI)  :  :  b  :  a. 

9.  Any  semi-diameter  is  a  mean  proportional  b(-tu:een  the  dir- 
tances  from  the  center  to  the  points  vhere  it  meets  the  ordinate  of 
any  point  and  the  tangent  at  that  point. 

For  the  point  p  (Fig.  82)  we  have 

p  =  cos  ^  .  u  +  sin  (^  .  /3. 
Also 

OT  =  a'OP  =  OQ  +  QT 

=  .t(cos ^  .  a  +  sin c/)  .  /3) 

=  cos</)'  .  a  +  sin  (/)'  .  ft  +  t{  —  sin  <;i'  .  a  +  eos^'  .  ft) . 

Eliminating  ;,  ^ 

cos(<^—  <^') 
1 


cos  ( <j()  —  <^  ) 

But 

ON  =  a''OP  =  OQ  +  QN 

=  a;'(cos<^  .  a  +  sin<^  .  ft) 

=  cosc^'.a  +  s'mcfi' .  ft  -f  /'(  —  sin<^  .u  +  COSe^  .ft). 

Eliminating  t', 

x'  =  C'0H{4>  —  4^')-, 
or 

OK  =  cos(<^  —  <^')op  ; 

.-.    ON  .  OT  =  op'-'. 


194  QUATERNIONS. 

10.    To  find  the  length  of  the  curve. 

With  the  notation  of  F^x.   6,  Art.  86,  we  obtain,  from  Eq. 

p'=  —  sin  <^  .  a  +  cos  ^  .  j3, 
Tp'=  V(a-  — 6-)sin-<^4-^r, 

.s  —  So  =  I  V(a"  —  6-)sin-c)l>  +  6-, 

which  involves  elliptic  functions.     If  a  =  Z>,  we  have,  for  the 

circle,  s  —  s^^=  \  r  =  r(^  —  <^u) . 
'■'o 
P'roni  Eq.  (220),  Ave  obtain 

p'=  a  —  m5(a-  —  x^)  -ix(3, 


T  '—      1  _i_      *'*'      ..2  f'  1      f''-'«" 

'        a-  —  ar  v«  —  x-  >  cr 


J,   ya-  —  X-  ^         a- 


which  may  be  expanded  and  integrated  ;  giving  for  the  entire 

curve 

_; S^ 3.3.  o  e'^        _      ^ 

2.2      2  .  2  .  4  .  -4      2  .  2  .  4  .  4  .  G  .  G       ^'  ^' 

a  converging  series.     If  e  =  0,  we  have,  for  the  circle,  2-^r. 
11.    To  find  the  area  of  the  ellipse. 
With  the  notation  of  Ex.  7,  Art.  8G, 

TVpp'=  TV(cos  (/)  .  a  +  sin  <^  .  /5)  (  -  sin  <^  .  a  +  cos  <^  .  /?) 
=  TV(co.s-</>  .  up  -  sin-<^  .  /3a)  =  TVa/3  ; 

A 

or,  since  aft  =  90°  ^_ 

i-f  ^TVpp'=l-7ra6. 

The  whole  area  is  therefore  iruh. 


APPLICATIONS   TO   LOCI.  19,'j 

90.   The   Hyperbola. 

1.    Let  a  and  (i  be  unit  vectors  parallel  to  the  a8jTnpU)tes. 
Then,  from  the  equation, 

a-  -\-  U- 

we  have,  for  the  equation  of  the  hyperbola, 

P  =  xa  +  '^(3 {22:,): 

or,  if  a  and  (3  are  given  vectors  parallel  to  the  asymptotes. 


P  =  <a+| (-i-^O) 


or,  again,  in  terms  of  the  eccentnc  angle, 

p  =  sec<^  .  a  +  tan(^  . /3  .      .      .      .       {'227). 

2.  The  equation  of  the  tangent,  obtained  as  usual,  is  from 
Eq.  (226), 


p^ta+'j+xUa-'j).       .       .       .        (228), 


where  ia  —  -  is  a  vector  along  the  tangent. 


91.    Examples  o)i  (he  II)jperhoJa . 


1.  If,  n-hen  the  hyperbola  is  referred  to  its  asymptotcH,  one 
diagonal  of  a  jyarallelogram  whose  sides  are  the  coordinates  is 
the  radius  vector,  the  other  diagonal  is  x>o,vcdlel  to  the  tangent. 

If  (Fig.  7'J)  cx  =  ^a,  \i'  =  -.  then 

cv  —  ta+^,     qx=ta—-; 


196 


QUATERNIONS. 


but  ?a  —  "  is  parallel  to  the  tangent  at  p  (Art.  90).     ta  -f--^  and 
ta  —^  are  evidentl}-  conjugate  semi-diameters. 

Fig.  79. 


2.  A  diameter  bisects  all  chords  parallel  to  the  tangent  at  its 
vertex. 

Let  (Fig.  79)  cr  be  the  diameter,  t  corresponding  to  the 
point  p.  The  tangent  at  p  is  parallel  to  ta  —!~  and  cp  =  ^a  -f-  ^. 
p'p"  being  the  parallel  chord, 

CP'=  CO  +  OP'=  xUa  +- )  4-  II  [t^  —      )• 

Also,  if  t'  correspond  to  pJ 


CP  =  ta  -)-^ 


••    {x  +  v) 


t==t'     ^iZLi/^1, 


t  V 


x^-f=.\. 


APPLICATIONS    TO    LOCI.  197 

Iloiur,  lor  uvcry  point,  as  o,  dctcM-iniued  liy  .»',  tluTO  tire  two 
points  !•'  and  r",  dctcrniined  by  the  two  corresponding  valnes 
of  y,  which  are  equal  with  opposite  signs. 

3.  The  tanfjent  at  r,  to  the  cohjurjute  hyperbola  is  parallel  to 
CP  (Fig.  7t)). 

4.  The  jwrtion  of  the  tangent  limited  by  the  asymptotes  is 
bisected  at  the  point  of  contact. 

5.  If,  froni  the  point  d  (Fig.  79),  tvhere  the  tangent  at  p  meets 
the  asymptote,  nx  be  drawn  parallel  to  the  other  asymptote,  then 
the  portion  of  pn  produced,  ivhich  is  limited  by  the  asymptotes,  is 
trisected  at  v  and  n. 

We  have 

ex 


2ta  +  xlB  =  t'a+l^,  =  2ta  +  ^^, 
P. 


CP  =  ?tt  + 


PX  =  ex  —  CP  =  ta  —  ^—, 

it 


and  the  equation  of  ss'  is 

whence,  for  the  points  s,  s,  i 

6.    The  intercepts  of  the  secant  between  the  kyperboh(  and  its 
asymptotes  are  equal. 

The  vector  along  the  tangent  parallel  to  the  secant  is  ta  —  -. 
Hence  (Fig.  79)  ^ 


H} 


Ck'  =  Za  =  .r  ^a  +f)  +  a 

Ck"  =  ^'^=  ..■  (ta  +  f )  +  llita  -  ^ 

but  op"  =  OP'  (Ex.  2),  and  therefore  p"k"=  p'u.' 


198  QUATEENIONS. 

7.  If  through  any  iwint  p"  (Fig.  79)  a  line  R"r'R'  he  (hxnvn 
in  any  direction,  meeting  the  asymptotes  in  r"  and  r',  then 

p"r"  .  p'r'  =  pd'-. 

8.  If  through  v\  p"  (Fig.  79)  lines  he  draivn  parallel  to  the 
asymptotes,  forming  a  parallelogram  ofivhich  p'p"  is  one  diagonal, 
the  other  diagonal  ivill  pass  through  the  center. 

The  vector  from  c  to  the  farther  extremity'  of  tlic  required 
diagonal  is 

But  t"a-^~  is  the  vector  from  c  to  the  other  extremitj'  of  the 
required  diagonal. 

9.  If  the  tangent  at  any  point  p  meet  the  transverse  axis  in  t, 
and  PN  he  the  ordinate  of  the  pioint  p  ;  then 

CT  .  CN  =  a-, 

c  being  the  center  and  a  the  semi-transverse  axis. 
From  Eq.  (227),  substituting  in  ct  =  cp  +  pt, 

xsec(j> .  a  =  sec^  .  a  +  tan^  .  (3  +  y/(tan<^  sec<^  .  a  +  sec-<^  .  /?) ; 

._      1 

sec^^ 
and 

CT  .  CN  =  (x  sec  (f>  .  a)  (sec  ^  ,  a)  =  a*, 
or 

CT  ,  CN  =  «-. 

10.  If  the  tangent  at  any  point  p  meet  the  conjngcde  axis  at  t,' 
and  pn'  he  the  ordinate  to  the  conjugate  axis,  then 

ct'  .  cn'  =  Ir, 
c  being  the  center  and  h  the  semi-conjugate  axis. 


UNiVERSITY 


V^L 


APPLICATIONS    TO    LOCI.  199 

92.  Tho  pivc('(liii;j;  ('x:un[)l('s  on  the  ooiiic  sections  involve 
directly  the  Cartesian  fonns.  A  method  will  now  bo  brietlv 
indicated  peculiar  to  Quaternion  analysis  and  independent  of 
these  forms. 

1.  The  general  form  of  an  equation  of  the  first  degree,  or  as 
it  may  be  called  from  analogy,  a  linear  ecjuation  in  quaternions,  is 

aqh  -\-  a'qh'-]-  a"qb"+ =  c, 

or 

'S.aqb  —  c,  (a) 

in  which  q  is  an  unknown  quaternion,  entering  once,  as  a  factor 

onl^-,  in  each  term,  and  a,  h,  a',  h', o  arc  given  quaternions. 

It  ma^'  evidently  be  written 

%^aqb  +  %yaqh  =  Sc  +  Vc, 
whence 

^^aqh  =  Sc,  {h) 

2Vag6=V&.  (c) 

But 

^aqh  =  ^qha  =  Sf/S6o  +  S  .  V?V6o, 
and 

yaqh  =  V(Sa  +  Va)  (Sg  +  Vry)  (S/>  +  yh) 

=  Y  .  Sry  (Sa  +  V«)  (S/j  +  ^h) 

+  y  (SaYr/S&  +  SaVryV^  +  VoY^yS^  +  YaYryY?;) 
=  S?Ya6  +  Y(SoS^  -  SaY6  +  S6Ya)  Yry 

+  Y  .  YaY(?Y6  +  Y  .  YaY6Yg  -  Y  .  ^ciShyq 
[Eq.  (1 1 G )]   =  SgYaZ>  +  Y(SaS&  -  SaY&  +  S6Ya  -  YaY&)  Yry 

+  2YaS  .  YryY/> 
=  Sr^YaZ>  +  Y  .  (((K6)Y7  +  2  YaS  .  YfyY?^. 

We  have  therefore,  from  (6)  and  ('•), 

Sc  =  S^2S6a  +  S  .  Y^SY^a, 

Yc  =  ^qViah  +  2Y  .  a  (K/y)  Yr/  +  2  2Y«S  .  YryY/). 

or,  writing 

2Sa6  =  d,     2Ya?>  =  6,     2Y6a  =  8,'     Sg  =  ?f,     Yr/  =  p, 


200  QUATERNIONS. 

we  obtain 

Sc'=  icd  +  Spo', 

\c  =  wS  +  2V  .  a{Kb)p  4-  2 2VaS  .  pYb. 

We  may  now  eliminate  iv  between  tliese  equations,  obtaining 

Vc  .  d  -  Sc  .  8  =  dSVa  {Kb)p  -  8SpS'+  d  2  2VaS  .  pYb 

which  involves  only  the  vector  of  the  unknown  quaternion  rj,  and 
which,  since  V  and  2  are  commutative,  ma^-  be  written  under 

the  ofeneral  form 

y  =  V/-p  +  2,5Sap, 

in  which  y,  a,  a,  ,  ^,  (3',  are  known  vectors,  r  a  known 

quaternion,  but  p  an  unknown  vector.  This  equation  is  the 
general  form  of  a  linear  vector  equation.  The  second  member, 
being  a  linear  function  of  p,  may  be  written 

Vrp  +  2/?Sap  =  <^,o  =  y     ....     (229), 

where  (^p  designates  any  linear  function  of  p.  If  we  define  the 
inverse  function  (^~^  In'  the  equation 

•••  P=  0"'7, 
the  determination  of  p  is  made  to  depend  upon  that  of  ^"'. 

2.  Without  entering  upon  the  solution  of  linear  equations,  it 
is  evident  on  inspection  that  the  function  <^  is  distributive  as 
regards  addition,  so  that 

4>{p  +  p'+ )  =  </,p  +  <^p'+     .     .     .     (230). 

Also  that,  a  being  an}'  scalar, 

ijiap  =  axf>p (231). 

and 

d(f>p  =  <f>dp (232). 

3.  Furthermore,  if  we  oijerate  upon  the  form 

<fip  —  2/8Sa/3  4-  \rp 


A1'1>LICAT10NS   TO    LOCI.  201 

with  S  .  rr  X  ,  o-  beiiiii"  any  vc'c-Un-  wliatevcr. 

Sa-<^/3  =  2S(rr/?Sri/j)  +  S(r(  V/'p)  . 

But 

S((r|8Sap)  =  SrrySSap  =  S;jaS/3.r  =  S(paS^rr)  , 

Hia-Yrp)  =  S[o-V(Sr  +  VOp]  =  S/'S-rp  +  Srr(yy)p 
=  HrSpa  -  Sp(V?-)o-  =  S[pV(K/-)(r]. 

Iloiiee,  if  we  designate  by  </)'o-, 

a  new  linear  funetion  ditiering  from  (/>  by  the  interchange  of  the 
letters  a  and  /?,  and  Kr  for  r,  we  shall  have,  whatever  the  vectors 
p  and  cr, 

S(rr(/>p)=S(p</)'(r). 

Functions,  which,  like  </>  and  <^;  enjoy  this  property,  are  called 
conjugate  functions.  The  function  </>  is  said  to  be  self-conjugate, 
that  is,  equal  to  its  conjugate  (f),  when  for  any  vectors  p,  o-, 

SiT<^p  =  Sp(^fr. 

93.  In  accordance  with  Boscovich's  definition,  a  conic  sec- 
tion is  the  locus  of  a  point  so  moving  that  the  ratio  of  its  dis- 
tances from  a  fixed  point  and  a  fixed  right  line  is  constant. 

1.  Let  F  (Fig.  80)  be  the  fixed  point  or  Fig  so. 

focus,  L>o  the  fixed  line  or  directrix,  and  p 

anv  i)oint  such  that  —  =  p,  the  constant  ratio 

Di- 
or eccentricity.     Draw  fo  perpendicular  to  ^ 
the  directrix,  and  let  Fo  =  a,  OD=yy,  vv  =  xx 
and  FP  =  p.     By  definition. 

To 


T(p.,) 

p-  =  e-x-a^. 

p  +  .)•  •.  =  a  -f-  ?/y. 


(a) 


202  QUATERNIONS. 

Operating  with  S  .  a  X  ,  we  have,  since  Say  =  0, 

Sap  +  Xa-  =  a", 

or 

X'a*  =  (a'  —  Sap)^. 


Substitnting  in  (a) 


aV-  =  e-(a2-Sap)-        ....      (233). 


in  which  e  may  be  less,  greater  than,  or  equal  to  unity,  corre- 
sponding to  the  eUipse,  hyperbola  and  parabola. 


Fig.  81. 


2.    For  the  ellipse.   Fig.  81,  putting   p  =  xa  for  the  points 
A  and  A^  we  have 


X  = and     a;  = •» 

1  4-e  1  — e 


or,  since  p  =  xa  =  a-FO, 


FA  = FO, 

1+e 


Fa'=  FO, 

1  -e 


whence 


and  therefore 


aa'  =  2  a  ^ 


2e 


l-e^ 
1  -(^- 


AI'PLICATIONS    TO    LOCI.  203 

Avliicli  runiisli  tlu'  well-known  proiK'rtics  of  tin-  ellipse, 

i.-A'=a(l  -\-e), 

OF  =  (le, 

1  -e 
AO  = «, 


3.    Changing  the  orighi  to  the  center  of  the  curve,  let  cf  =  a  ; 


3     and    p  =  p— a,    a=i  — 
1  -e-'   , 


then   cp  =  p    and    p  =  p'-  a,    a'  =  (  — -^,  -  — -^  )  a  ;    whence 


-  a!     Substituting  these  values  of  p  and  a  ii 
a-p-  =  e-(a"  —  Sap)", 
remembering  that  a'-  =  —  a-e-,  we  obtain 

a-p"  +  (Ha'p')'  =  -(('(l-e-), 
or,  dropping  the  accents,  c  being  the  initial  point, 

.rp'  +  (S-p)'  =  -^''(l-e')       •     •     •     ("^34), 

the  equation  of  the  ellipse  in  terms  of  the  major  axis  with  the 
origin  at  the  center.     If  p  coincides  with  the  axes,  Tp  =  a  or  &, 

as  it  should. 

4.  Equation  (234)  ma}"  be  deduced  directly  from  Newton's 
defmition,  thus:  let  cf  =  a  (Fig.  81)  as  before,  f  and  f'  being 
the  foci,  and  ci>  =  p.     Then 

FF  =  p  —  a,      f'f  =  p-\-a^ 

and,  by  defniition, 

FP  +  F'r  =  2a 

as  lines  ;  or 

T(p-a)+T(p  +  a)=2a, 


204  QUATERNIONS. 

a  being  the  semi-major  axis.     Whence 

V— (p  — a)-=  2a  —  V— (p  +  a)- 

Squaring 


-  p-  +  2  Spa  -  a-  =  4  rt-  -  4  rt  V  -  (p  +  u )  -  -  p2  _  2  Spa  -  a^ 

Spa  —  a- =  —  «  V— (p  +  a)^ 
Squaring  again 

(Spa)-  -  2  a- Spa  +  a*  =  -  a-(p-  +  2  Spa  -f  a")  , 
a-p^  +  (Spa)^  =  —a*  —  a- a-, 
or,  as  before, 

a-p-  +  (Spa)-  =  -a\l  —  e^). 

94.    1 .  The  equation  of  the  clhpse 

a-p-  +  (Sap)-=  —  a''(l  —  e^) 
ma}'  be  put  under  the  form 

or,  in  tlie  notation  of  Art.  92,  writing 

«^p  +  aSap , 

the  equation  of  the  ellipse  becomes 

Spc^P=l (235). 

2.    By  inspection  of  the  vahie  of  (f>p  it  is  seen  that,  wlien  p 
comcides  with  either  axis,  p  and  <^p  coincide. 
Operating  on  ^p  with  S  .  o-  X  ,  we  have 

■    ^     ,  a^STp  +  S.TaSap 

a*(l  — €-) 


ArPLICATIONS    TO    LOCI.  206 


a'  (T  +  aSarr 
) 
a-  Spo-  4-  S.oaSao- 


operating  on  <^o-  = — ^  with  S  .  />  x  ,  wo  liuve 


hence 

Sp</>o- =  Scr^/) (230), 

and  ^  is  self-conjugate. 

3.  Differentiating  Ivjiiatiou  (23.")),  we  iiave 

Sclp<f>p  +  Spdcfip  =  0, 

Mp<lip  -+-  Spcf^dp  =  0,  [Eq.  ( ■2:V> )  ] 

^p<f)dp  -f-  Sp4>dp  =  2  S,Q^r/p  =  0.  [ Kq .  ( 23(; )  ] 

If  TT  l)c  a  vector  to  any  point  of  the  tangent  line, 

TT  ^  p  -\-  xdp, 
whence 

S/j^(7r  — /j)=S(7r  — /j)(^p  =  0,  («) 

or 

Sttc^P  =  Sp<^p  ^  S,O07r  =:  1     ....       (237) 

is  the  equation  of  the  tangent  line. 

From  {(()  we  see  that  (f)p  is  a  vector  parallel  to  the  nornial  at 
the  point  of  contact,  being  parallel  to  p  only  when  p  coincides 
with  the  axes,  as  already  remarked. 

4.  To  transform  the  preceding  cqnntions  into  the  usual  Car- 
tesian forms,  let  i  he  a  unit  vector  along  ca  (Fig.  81),  and  j  a 
unit  vector  perpendiculai*  to  it.  If  the  coordinates  of  p  are  x 
and  ?/,  then,  since  a  =  ci\ 

P  =  xi  +  W^ 
and 

_      a-p  -f  aSap  _  _  a-(xi  -f-  ///)  +  aeiS  .  a(n'(xi  -f- ;//) 
'f'P~~  a^(l-e-)  "  a'{l-(r)  ' 


206  QUATEllNIONS. 


or,  since  1  - 

.-.   Sp4>p  = 

=  l=~i 

-i  .  {xi  -\ 

and 

')(^+l> 


a-y-  +  Irx-  =  a- Jr. 
Again,  if  x'  and  y'  are  tlie  coordinates  of  a  point  in  the  tangent, 
TT  =  x'i  +  //;/  ; 

and 

a-  yy'  +  &-  .^.u*'  =  ((-  Ir. 

Tlie  abo's-e  applies  to  tlie  hyperbola  wlien  e  >  1 ,  that  is,  when 

1  —e-  — -,  giving  the  corresponding  equations 

cr  " 

a-y-  —  Irx-    =  —  (rb'-, 
a^yy'—  b'-xx'  =  —  crb-. 

95.   Examples. 

1 .     To  find  the  locus  of  the  middle  points  of  j)arcdlel  chords. 

Let /S  be  a  vector  along  one  of  tlie  chords,  as  kq  (Fig.  82), 
the  length  of  the  chord  being  2?/,  and  let  y  be  the  vector  to  its 
middle  point ;  then 

l>  =  y  +  Iff^     '^"^^     P  =  7  -  2/^ 

are  vectors  to  points  of  the  ellipse,  and 

S(y+///3)<^(7+.V)8)  =  l, 
H{y-yfS)4>(y-y/3)  =  A; 

whence,  expanding,  snbtracting,  and  api)lying  Equation  (23(;), 

Sy4>/3=0, 


APPLICATIONS    TO    LOCI. 


207 


till'  ('(juation  of  ji  straiulit  liiu'  tlirouuli  tlu'  origin.  Since  cf>ft 
i.s  parallel  to  the  normal  at  the  extremity  of  a  diameter  parallel 
to  /3,  tlie  locus  is  the  diameter  parallel  to  the  tangent  iit  tliat 
point. 


2.  Equation  of  condition  for  conjitgate  diameters. 

Denote  the  diameter  <.)i'  (Fig.  82)  of  the  preceding  problem, 
bisecting  all  chords  parallel  to  ^,  by  a.     Then 

Sa</,/S  =  0, 
or 

SySc^a  ^  0. 

Ill  the  latter,  yS  is  perpendicular  to  the  normal  <i>a  at  the  ex- 
tremity of  a,  and  is  therefore  ])arallel  to  the  tangent  at  that 
point;  hence  this  is  the  equation  of  the  diameter  bisecting  all 
chords  parallel  to  a.  Therefore,  diameters  which  satisfy  the 
e(iuation  Sa<^/?  =  0  are  conjugate  diameters. 

3.  Siqyplementar>i  chords. 

Let  vr'  (Fig.  .S2)  and  dd'  be  conjugate  diameters,  and  the 
chords  pi>,  vd'  I)e  drawn.     Then,  with  the  above  notation, 


and 


l)V  =  a  —  /3, 
S(a  + /?)(/.(a  -  |8)  =  S(«  + /3)  ( 0.  -  <^/3) 


208  QUATERNIONS. 

But 

Sa</>a  =  1  ,       S/3</,/3  =  1 .      Sa0/3  =  S,8c}>a  ; 
.-.     8(a+y3)<^(a-/3)  =  0. 

Hence,  if  dp  is  parallel  to  a  diameter,  pd'  is  parallel  to  its 
conjugate. 

4.  If  two  tangents  be  drawn  to  the  ellipse,  the  diameter  parallel 
to  the  chord  of  contact  and  the  diameter  through  the  intersection 
of  the  tangents  are  conjugate. 


Let  TQ  (Fig.  82)  and  tr  he  the  tangents  at  the  extremities  of 
the  chord  parallel  to  /S,  and  ox  =  tt.     Then 

oq  =  XuL  +  ?//3,     OK  =  xa  +  i/'(S. 
From  the  eqnation  of  the  tangent  Sttc^/j  =  1 ,  we  have 

S7rc/)(.l-a  +  //^)=l, 
S7r<^(.i-a +  //'/?)=  1. 

^  Expanding  and  subtracting 

H7rct>(S  =  0. 

Hence,  Ex.  2,  tt  and  /5,  or  ov  and  od,  are  conjugate.  The 
locus  of  T  for  parallel  chords  is  the  diameter  conjugate  to  the 
chord  through  the  center. 


APPLICATIONS    TO   LOCI.  200 

5,  If  Qoq'  (F\g.  82)  be  a  diameter  and  q\i  a  chord  of  cotiUirt, 
then  is  q'k  parallel  to  ot. 

KQ  being  panillol  to  /3,  and  oq'  =  —  oq,  we  have 
KQ  =  2  ^'//?,     uq'  =  1/(3  —  xa  —  xa  —  y(i  ; 

whence,  clirec%  rq'  =  —  2.ra  ;  as  also  Srqc^uq'  =  0,  kq  and  itc/ 
being  supplementary  ehords. 

0.  The  points  in  ichich  amj  two  parallel  tangents  as  </t'  qt 
(Fig.  82)  are  intersected  bij  a  third  tangent,  as  ttJ  lie  on  conju- 
gate diameters. 

The  equation  of  kt'  is  S~cfiji  =  \,  and  that  of  q't'  is  Sn-'<^/j'=l. 
For  the  point  tJ  tt  =  tt'  ;  wiience,  by  subtraction, 

S;rc^(p-//)-0. 

7.  Chord  of  contact. 

The  equation  of  the  tangent, 

Hp(f)Tr  =  1 , 

is  linear,  and  satislied  for  both  Q  and  r.     Hence,  writing  a  for  p 
as  the  variable  vector,  tt  being  constant, 

S:Tcf>7r  =  1 

is  the  equation  of  the  chord  of  contact. 

8.  To  Jind  the  locus  of  t  for  all  chords  through  a  Ji.i-cd  point 
(Fig.  82). 

Let  s  be  a  fixed  point  of  the  chord  rq,  so  that  os  =  .r  =  (t 
constant.     Then 

Scr<f>Tr  =  Srrcfia-  ==■  1 , 

a  right  line  perpendicular  to  <^3-,  or  parallel  to  the  tangent  at  the 
extremity  of  os,  and  the  locus  of  x  for  all  chords  through  s. 


210 


QUATERNIONS. 


9.  Any  semi-diameter  is  a  mean  proiwrtional  between  the  dis- 
tances from  the  center  to  the  points  where  it  meets  the  ordinate  of 
any  point  and  the  tangent  at  that  x>oint. 


on  {¥\g.  82)  and  oi>  being  still  represented  by  /3  Jind  a,  let 
OT  =  x'a  and  oq  =  p  =  x-a  +  yft.  Then  from  the  equation  of  the 
tangent,  S;r^p  =  1 ,  we  obtain 

Sx-'a</)(.m +  ?//?)=  1; 

whence,  since  Sa^^=  0, 

xx'^ci^a=  1, 
or 

XX' =\: 


.'.    Xa  .  X'a=  a^, 

or 

ON  .  OT  =  OP-. 

10.    I/dd'  (Fig.  82)  and  pp'  are  conjugate  diameters,  then  an 
PI)  and  pd'  2'>ro20ortional  to  the  diameters  parallel  to  them. 

With  the  same  notation 


whence 


OE  =  m  (a  —  /5)  ,       OF  =  n  (a  -f-  yS)  , 


APPLICATIONS    TO    LOCI. 


2il 


^  and 


From  the  equation  of  the  ellipse 

^m{a-^)cl>m(,i-(3)=l,  (a) 

Sn(a  +  j8)</)U(a+/3)=l.  (l) 

Now,  from  (a),  since  ^/3(j>(3  =  Sa<^a  =  1  and  S/3(f>a  =  iya<t>j3  =  0, 

2  m-  =  1 . 
Similarly,  from  (^), 

■2  H-  =  1  ; 

.'.   m  =  II. 
Also 

m-:r.'l'::T(a-/3)  :  T(a  +  /3) 

::Tm(a-/3)  :  T)((a  +  /?) 
:  :  OE  :  ov. 

11.  TJie  diameters  along  the  diagonah  of  the  parallelogram  on 
the  axes  are  conjugate;  and  the  same  is  true  of  diameters  along 
the  diagonals  of  any  parallelogram  tvhose  sides  are  the  tangents  at 
the  extremities  of  conjugate  diameters. 

12.  Diameters  parallel  to  the  sides  of  an  inscribed  parallelo- 
gram are  conjugate. 

Fig.  S3. 


Let  the  sides  of  the  parallelo^-am  (Fig.  83)  be 
pp'  =  a,     FQ  =  P, 


and  let 
Then 


OP  =  p,       OQ  =  p. 
op'  =  p  +  a,      OQ'  =  p'  -f-  a,      p'  —  p=:  ff. 


212 


QUATERNIONS. 


From  the  equation  of  the  ellii^se,  Sp^/3=  1 ,  we  have  for  q'  and  p' 

S(p'4-a)</>(p'+a)=l,  I 

S(p+a)c/,(p+a)=l; 

whence,  since  Sp0p  =  ^p'  i^p'  =  l^ 

2  Sa<^p' +  Sa</>a  =  0, 
2  fiacfyp  +  Sac^a  =0. 

S,o  <^a  —  Sp<^a   =  0, 

S(p'-p)c/)a  =  S/5c^a  =  0. 


Subtractina; 


13.  The  rectangle  of  the  2)erpendiculars  from  the  foci  on  the 
tangent  is  constant,  and  equal  to  the  square  of  the  semi-conjugate 
axis. 

Fis-.  S3. 


Let  the  tangent  be  drawn  at  r  (Fig,  83)  and  or  =  p.  Then 
^p  is  parallel  to  the  normal  at  r,  that  is,  to  the  perpendiculars 
FD,  f'dI     Hence,  of  being  a, 

Od'=  x'  (ftp  —  a, 
OD  ^  a  +  X<f>p, 

which,  since  d  and  d'  are  on  the  tangent,  in  S7r<^p  =  1  give 

S (x'  (f)p  —  a)  (fip  =  1 , 
S(a  +  X(t>p)(l>p=  1, 

or 

x'{<f>pY=l+Sa<f>p, 
X  {(f>p)'  =1  —  Sac^p  ; 


whence 


and 


AIM'LU'ATIONS   TO   LOCI.  213 

TXchn    =  Kl)    =  T -— ^, 

.       W 

„ ,..,       ^l-(Sa<^p)2 


But 

/      g^p  +  ftSapV,  ft-(a-p-)  +  2a^(Sap)-  +  a-(Sap)^ 

or,  substituting  a-p'  from  Equation  (234)  and  a-  =  — a-e'-, 

«"(1  -e-)* 
Also 

1        .c   ^   X'        1        prSrxo  +  a^S:xp7__  a^-(Sap)^ 

1  -  (Sa<^p)-  =  1    -  [        «4(l_g2«>       J   -  ^^4 

,    ,         a''-(Sap)2  o'-'(l-e-)         ,^^        „,       ,, 

14.    r//e  ./bo«  o/  </ie  perpendicular  from  the  focus  on  the  tanr/ent 
is  in  the  circumference  of  the  circle  described  on  the  major  axis. 

To  prove  this  we  have  to  sliow  that  tlie  line  on  (BMg.  83)  is 
equal  to  a.     Now 

01>  =  a  +  Xcf)p 

<J3p{\  —  Sa<^p) 
=  "+— (^^ 

from  the  preceding  exam][)le.     Ilonee 

^       ^,  „       2Sa<^p(l  -SacAo)    ,    (l-Sacf>py 

„      l-(Sa<^p)2  ,  ,  .  a'-(^apra\\-e') 

^'^'^       ii^pY       =  - '''''  +         a^         (Sap)^-  a^ 
=  -a2e2_«2(l  -e-)  =  -a^; 


214  QUATERNIONS. 


The  Parabola. 

96.    1.    Resuming   Equation    (233)    and   making   e  =  l,    the 
equation  of  the  parabola  is 

aV^-(a2-Sa/,)2 (238), 

which  may  be  written 

p-  +  2Sap-a--(Sap)-^^^ 


in  which,  if  we  put 

,  p  —  tt"'  Sap 

W=  -o 5 

a 

we  have  for  the  equation  of  the  parabola 

S,Q(</)p  +  2a-')  =  l (230), 

and,  as  in  the  case  of  the  ellipse, 

So-<^p  =  Sp</>(r •    .     (240). 

Operating  on  <^p  b\-  S  .  a  x  ,  we  obtain 

Sa<^p  =  0 (241); 

hence,  (ftp  is  a  xierpencUcidar  to  the  axis. 
Operating  on  ^p  b^-  S  .  p  x 

SpcAp  =  ^^--y-^>^=a^(<Ap)^.      .      .     (242). 

2.    Differentiating  Equation  (239),  we  have 

2Sp<^dp  +  2Sr/pa-^  =  0. 
For  anv  point  of  the  tangent  line  to  Avhich  the  vector  is  tt, 
TT  =  p  +  xdp^ 


APPLICATIONS    TO    LOCI.  215 

from  which,  substituting  (Ip  in  tlic  alxn'O, 

Sp<^(7r-p)      +S(7r-p)a-i  =  (), 

S(p^7r  — p<^p  +  7ra-'— pa"')  =  0  ;  (o) 

or,  since  Sp<^p  =  1  -  -iSiu-'  [Eq.  (2;31))]. 

Sttc^/j  —1+2  S,ofi-'  +  Sttu-'  —  Spa-'  =  0. 

whence 

S7r(<^P  +  a-')  +  Spa-'=l   .       .       .       .      (243). 
the  equation  of  tlie  tniiucnt  line. 

3.  From  (a)  we  obtain 

S(7r-p)(<^p  +  a-')  =  0; 

or,  since  tt  —  p  is  a  vector  along  the  tangent, 

<^p  +  a-' 

is  in  the  direction  of  the  nnrmal. 

4.  If  o-  be  a  sector  to  an}-  point  of  the  normal,  the  equation 
of  the  normal  will  be 

(r  =  p+.r(<^p  +  a-') (244). 

5.  The   Cartesian    form    of   Equation   (239)    is   ol)tained    b\- 
making 

p  =  .(•/  +  ]ij.     a  =  vo  (Fig.  80)  =  —jyi ; 

xj^i 

■xi  4-  /// 

____£._    yj. 

.  • .    9p  = :; = ., ' 

-p-  ir 

whence.  Equation  (239)  becomes 

jr       p 
•••  r=27u-+ir, 
the  equation  of  the  parabola  referred  to  the  focus. 


216  QUATERNIONS. 

97.   Examples. 

1.    The  subtauyent  is  bisected  at  the  vertex. 

Fig.  84. 


(«) 


We  have  (Fig.  84)  ft  =  .ra,  which  in  the  equation  of  tlie  tangent 

S7r(<^p  +  a-i)  +  Spa-'  =  l 
gives 

Sa;a(</>p  +  a-')  +  Sa-'p  =  l. 

But  Sttt^p  =  0  ;  hence 
multiphing  b^-  a 

Xa  +  aHa~^  p  =  a, 

{x  —  ^)a  =  a  —  ^a  —  aSa~^  p 
=  -^a  —  aSa~'p, 
AT  =  —  AF  —  aSa~^  p. 

But  the  vahie  of  (f>;)  gives 

u"  (f)p  =  p  —  a~  Hap  ; 


APPLICATIONS   TO    LOCI.  217 

ami,  since  (jip  is  a  vector  along  mi*  and  a~'Sap  a  vector  uloiii^'  km, 
from  /»=  FM  4-  MP  we  have 

lOI  =  a~'  Sap  =  aSa"^  p,  (6) 

Ml' =  «-()[>/.;  (c) 

. • .    AT  =  —  AK  —  FM  =  —  AM, 

or,  as  lines, 

AT  =  AM. 

2.  The  distances  from  the  forua  to  the  point  of  contact  and  the 
intersection  of  the  tamjent  icith  the  axis  are  erjnal. 

X  i.=  a  —  uSa-'p, 

or  (Fi<r.  ,S4), 

(KT)^=(.x-aSa-'p)'' 
=  (a  —  a~^Sap)' 
(a~  —  Sap)- 

[Eq.  CiSH)]  =p^- 

.  • .    KP  =  FT. 

3.  The  stthiiornial  is  constant  and  erjual  to  half  the  parameter. 

The  vector-normal   being    ^o -{- a~^    (Art.   96,   3),   we   have 
(Fig.  84) 

px  =  2(c^.  + a-'); 
but 

px  =  vy\  +  MX 

=  —  a-<^;j  +  a'a  ;  [Ex.   1 ,  (f) ] 

.-.    2;(^p  +  a~')  =  —  a-<^p +  .X'a, 
2  =  —  a-  =  .X'a-, 

or 

.r  =  —  1  ,       X(L  =  —  a  ; 

or,  the  distances  mx  and  fo  arc  equal,  and  the  subnormal  =  y^ 
a  constant. 

4.  The  perpendicidar  from  the  focus  on  the  tangent  intersects 
it  on  the  tangent  at  the  vertex,  and  aq  =  ^mp  (Fig.  84). 


218 


QUATEKNIONS. 


Since  (Ex.  2)  fp  =  ft  =  pd,  fd  is  perpendicular  to  pt  or  par- 
allel to  PN.     Otherwise  : 

NP  =  -  2(</)a  +  a-^)  =  a=(</>p  +  a"')  (Ex.  3) 

=  a-0p  +  a  =  MP  +  FO,  [Ex.   1,  (c)] 

a^(fip  +  a  =  FO  +  OD  =  FD. 

But 

^  FI>  =  FQ  =  I  u-^p  +  i  a 
=  |a-^p  +  FA  ; 
.*.    ^a-</)p  =  AQ  =  ^MP. 

5.  To  find  the  locus  of  the  intersection  of  the  perpe7idicvlar 
from  the  vertex  on  the  tangent  and  the  diameter  produced 
through  the  point  of  contact. 

Fig.  84. 


S            ] 

0 

Fy^ 

\ 
\ 

> 

ff\\ 

T                                O 

A 

V                      M                        N 

Let  FS  =  o-  (Fig.  84)   be  a  A'ector  to  a  point  of  the  locus. 
Then 

FS  =  FA  +  AS  =  FP  +  PS, 
cr  =  ^  tt  +  2  ((/)p  +  a->)  =  p  +  X-a. 


APPLICATIONS   TO   LOCI.  219 

Operating  uith  X  S  .  <^/j,  then,  since  Sd<^p  =  0  [Eq.  (-241)]. 
z{ct>py  =  S>p  =  u-(0;.)- ;  [K(i.  (212)] 

.•.    Z  =  a-, 

and 

o-  =  ^ a  +  a-(<^/j  +  a" ')  =  j;  a  +  u'-</)p, 

or 

(T  —  ^  tt  =  a-'<^p. 

Operating  with  X  S  .  a 

S(o--|a)a=0, 
Scra  =  -|(Ta)2, 

or  [Eq.  (l-'^O)].  the  locus  is  a  right  hne  perpendicuhir  to  the 
axis  and  |i>  distant  from  the  focus. 

6.  To  find  the  locus  of  the  intersection  of  the  tamjent  and  the 
perpendicidar  from'  the  vertex. 

If  the  origin  be  taken  at  the  vertex,  then  since  4>p  +  a"'  is  a 
vector  along  the  normal,  the  equation  of  the  locus  will  be 

,r  =  .T(</)p  +  a-').  (a) 

To  eliminate  x,  operate  with  S  .  a  x  Avhich  gives 

X  =  Saw,     whence     Sa"  V  =  —  '—. 
«- 

To  eliminate  p.  the  equation  of  the  tangent,  87r(<^/j  +  trM  + 
Soa~'  =  1 ,  for  the  new  origin  becomes 

S  ^TT  +  ^y  c^p  +  U-' )  +  Spa-' =  1 . 

or 

2 S7r<t>p  +  2  Sa-'-  +  2  Sa-'p  =  1 . 

Operating  on  (o)  with  x  S  .  ^p,  whence  S-^p  =  ,v(<^p)'-.  the 
precedmg  equation  becomes 

2.f(</)p)--^^  +  2Sa-'p=l.  ih) 


220  QUATERNIONS. 

Also  [Eq.  (242)]  "Sp^p  =  a-((/)p)-,  which,  in  the  equation  of 
the  parabola  Sp(</>p  +  2a~')  =  1,  gives 

a-(c^p)--l-2Sa-V=l.  (C) 

Whence,  from  {h)  and  (c) ,  l\y  subtraction, 

2  crx  +  a* 
But,  from  (a), 

(w)'  = ^ =  -  +  — • 

X-  X-      cr 

Equating  these  values  of  (0p)',  and  substituting  the  value  of  x, 

2  TT-SaTT  -  a-TT-  +  (SaTr)-  =  0, 

which  is  the  equation  of  the  locus  required.     To  transform  to 
Cartesian  coordinates,  make 

TT  =  xi  +  I/J,     a-nd     a  =  ai, 
whence 

TT-  =  —  (x-  +  ?/-)  ,      SaTT  =  —  rt.f ,      a-  =  —  a-, 
and 

x' 

X 

2 

the  equation  of  the  cissoid  to  the  circle  Avhose  diameter  is  the 
distance  from  the  vertex  to  the  directrix. 

7.  If  pp'  (Fig.   75)   be  a  focal  chord,  and  pa,  pa'  produced 
meet  the  directrix  in  d',  d,  then  will  pd  and  p'd'  he  parcdlel  to  af. 


Ad'=  —  X\V  =  AO  +  OD,' 

(a  \        a    , 


Operating  with  S  .  a  x 

a(a- —  2  Sap)  =  a-.  ((f) 


APPLICATIONS    TO    LOCI. 


221 


Fi,^'-  75  (biK). 


Now  Fr  =  p  and  Fr'=  — -^''^  ai"*^*  vectors  to  points  on   llu 
cm've,  and  honoe  satisfy  its  ('(nia- 
tion.     Whence  [Kq.  (•-^•"■'O] 


a-p-  =  (a"  —  'Sap)", 
x'-a-p-  =  (a-'  +  x'Sap)-  ; 


,   x"{a'  -  Sap)-  =  {a'+  x'HapY 

x'{a-  —  Sap)  =  a-  +  .r'Sap, 
.  • .    x'  (a-  —  2  Sap)  =  a". 

Hence,  comparing  with  (a), 


or,  the  sides  prodnced  of  the 
triangle  apf  are  cnt  propor- 
tionately, and  therefore  d'p'  is 
})arallel  to  af. 

8.  If,  icith  a  diameter  equal  to  three  times  the  focal  distance, 
a  circle  be  described  tcith  its  center  at  the  vertex,  the  common 
chord  bisects  the  line  Joining  the  focus  and  vertex. 

The  equation  of  the  curve  being 

ay  =  ("'-Sap)-,  (rt) 

that  of  the  circle  whose  center  is  a  (Fig.  75),  referred  to  f,  is 
of  the  form  [Eq.  (210)] 

T(p-y)   =T^, 
or,  by  condition, 


'(.-i)-: 


which,  in  (a),  gives 
which  is  the  proposition 


P~7,       —  TiT' 
P'  =  ^'^P  +  f  6  "'' 
iiap  =  — , 


222  QUATERNIONS. 

98.  The  Cycloid. 

1.  Let  a  and  /3  be  vectors  along  the  base  and  axis  of  the 
cycloid  and  T/?  =  Ta  =  r,  the  radius  of  the  generating  circle. 
Then,  for  an}-  point  p  of  the  cur^e, 

X  =  rO  —  r  sin  6  =  r{$  —  sin  6) , 
y  =  r   —rcos6  =  r{l  —  cofi6), 

and  the  eqnation  of  the  cycloid  is 

p  =  {e-  sin^)a  +  (1  -  cos^)/?. 

2.  Tlie  vector  along  the  tangent  is 

(1- cos  6)a  +  s'mO  .  13, 
and  the  equation  of  the  tangent  is 

7r  =  (^-sin^)a  +  (l-cos^)/3  +  <[(l-cos^)a  +  sin^  .  y8]. 

3.  The  vector  from  p  to  the  lower  extremit}-  of  the  vertical 
diameter  of  the  generating  circle  tlirough  p  is 

PC  =  —  (1  —  cos^)^  4- sin  ^  .  a, 

and,  from  the  above  expression,  for  the  vector-tangent  ft, 

S(pc  .  pt)  =  0  ; 

hence  pc  is  perpendicular  to  the  tangent,  or  the  normal  passes 
through  the  foot  of  the  vertical  diameter  of  the  generating  cir- 
cle for  the  point  to  which  the  normal  is  drawn,  and  the  tangent 
passes  through  the  other  extremity. 

4.  If,  through  p,  a  line  be  drawn  i)arallel  to  the  base, 
intersecting  the  central  generating  circle  in  q,  show  that 
PQ  =  r(7r  —  ^)  =  arc  QA,  a  being  the  upper  extremity  of  tlie 
axis. 


Al'I'LICATIONS   TO   LOCI.  223 

5.  AVith  the  notation  of  Ex.  G,  Art.  8G, 

p'  =(i_eos^)a+  sin^  .  /S, 
p'2  =  _  [(1  -  cos^)-  +  sin-^]?-, 


Tp'  =  r  Vl  -2  cose  +  cos-fy  +  siir^  =  r  V2-2cos^ 
s  -  So  =  ("2  r  siii4^^  =  [4  )•  cos|^]^_  =  8  r, 
the  length  of  the  entire  curve. 

G.  With  tlie  notation  of  Ex.  7,  Art.  86, 

TV/V=TV[(^-sin^)sin^  .  ay8  +  (I  -  cos^)-/3a] 
=  TV[(^  sin^  -  sin-^  -  (1- cos^)-]a/8 
=  )-2(^sin^  +  2cos^-2). 
A  -Ao  =  7^  fie  sine  +  2  cos  ^  -  2) 

=  r^(sin^-^cos^  +  2sin^-2^)T 

=  r^(3sin^-^cos^-2^)  T=3  7r?-2, 

the  whole  area  of  the  curve. 

99.   Elementary  Applications  to  Mechanics. 

1.  If  6  be  the  magnitude  of  an}'  force  acting  in  a  known  di- 
rection, the  force,  as  having  magnitude  and  direction,  may  be 
represented  by  the  vector  symbol  /8,  which  is  independent  of 
the  point  of  application  of  the  force.  In  order,  completely,  to 
define  the  force  Avith  reference  to  any  origin  o,  the  vector  OA  =  a, 
to  its  point  of  application  a,  must  also  be  given.  For  concur- 
ring forces,  whose  magnitudes  are  b[  b','  ,  we  have,  for  the 

resultant,  /8  =  2/8,'  which  is  true,  whc>ther  the  forces  are  compla- 
nar  or  not.  and  is  the  theorem  of  the  pob/gon  of  forces  extended. 
For  two  forces,  l3  =  fi'+(3";  whence  /8-'  =  fS"  +  /8"-  +  2  S/8'/8",  or 


224  QUATERNIONS. 

y^  =  &'2  4-  &"2  +  2h'h"  cos^,  which  is  the  theorem  of  the  parallelo- 
gram  of  forces.  For  any  number  of  concurring  forces,  the  con- 
dition of  equilibrium  will  be  2/?'=  0.  For  a  particle  constrained 
to  move  on  a  plane  curve  whose  equation  is  p=  (fi(t),  dp  being 
in  the  direction  of  the  tangent,  since  the  resultant  of  the  ex- 
traneous forces  must  be  normal  to  the  curve  for  equilibrium,  we 

have 

MpXl3'^Mpl3  =  0.  (a) 

2 .  If  oa'  =  a',  and  (3'  is  a  force  acting  at  a^  then  TVa'yS' =a'b'  sin 9 
is  the  numerical  value  of  the  moment  of  the  couple  j3'  at  a'  and 
—  /8'  at  o.  Representing,  as  usual,  the  couple  b}-  its  axis,  its 
vector  symbol  will  be  Va'/3!  If  — /3'  act  at  some  point  other 
than  the  origin,  as  c',  and  oc'=  y',  the  couple  will  be  denoted  by 
-V(a'— y')^.'  From  this  vector  representation  of  couples,  it  fol- 
lows that  their  composition  is  a  process  of  vector  addition ;  hence 
the  resultant  coup)le  is  SV(a'— y');8;  and,  for  equilibrium, 
2y(a'— y')^'=  0.  If  the  couples  are  in  the  same  or  parallel 
planes,  their  axes  are  parallel  and  T2  =  2T.  Since  a'— y^  is 
independent  of  the  origin,  the  moment  of  the  couple  is  the  same 
for  all  points.  Since  V(a'—  y')/^'=  Va'^'— Vy'/^J  the  moment  of 
a  couple  is  the  alr/ebroic  sum  of  the  moments  of  its  compoyient 
forces.  If  the  forces  are  concurring,  and  a'  is  the  vector  to 
their  common  point  of  application,  2Va'^'=  T2a'/3'=  Va'2/3'= 
Ya'^,  or  the  moment  of  the  residtant  about  any  point  is  the  sum 
of  the  moments  of  the  component  forces.  When  the  origin  is  on 
the  resultant,  a'  coincides  with  /3'  in  direction,  and  Ya'yS  =  0  ;  or 
the  algebraic  sum  of  the  moments  about  any  point  of  the  resultant 
is  zero.  If  a  single  force  /?'  acts  at  a'  we  may,  as  usual,  intro- 
duce two  equal  and  opposite  forces  at  the  origin,  or  at  any  other 
point  c',  and  thus  replace  (3'^.  by  /8'o  and  Va'^'  or  by  ^V-  ''^"c^ 
y  (a'_  y')^(  If  ^  be  a  unit  vector  along  any  axis  oz  through  the 
origin,  then  the  moment  of  13'  acting  at  aJ  with  reference  to  the 
axis  oz,  will  be  -  S^'a'^,  or  -  S  .  W'^'  I^  (^'  -'^"^l  C  are  in  the 
same  plane,  in  which  case  they  either  intersect  or  are  parallel : 
or,  if  the  axis  passes  through  a'  there  will  be  no  moment :  in 
these  cases,  a',  ft'  and  C  are  complanar,  and  —  S^'a'^  =  0. 


APPLICATIONS   TO   LOCT.  225 

3.  If  the  forces  arc  pnnillel,  their  resultant  (i  =  1[i'=  %b'\]ft' 
=  U/32^' ;  find,  therefore,  for  eqnilibriiiin,  2T/8'=  2//=  0.  The 
moment  of  a  force  with  reference  to  an}-  axis  oz  through  tlie 
origin  being  —  S^'a'C,  and  the  moment  of  the  resultant  !)eing 
equal  to  the  sum  of  the  moments  of  the  components,  wc  have 
S(3aC='S,Sf3'aX,  which,  f(M-  parallel  forces,  becomes  S(2/>' .  V(3  .  (tO 
=  S{V(3'^h'a'  .  0'  "^vhich,  being  true  for  any  axis,  is  satislied 
for  2//  .  a  =  5//a' ; 


which  is  hidependcnt  of  U/?,  and  hence  is  the  vector  to  the  cen- 
ter of  ixiralM  forces.  AVheu  26'=  0,  the  above  equations  give 
^  =  0  and  a  =  00,  the  system  reducing  to  a  couple.  For  a  sys- 
tem of  particles  whose  weights  are  ic\  v','  ,  we  have  the  vec- 

tor  to  the   center  of  qravity  a  =  ^^—r-     From   this  equation, 

■     '  zio' 

2?c''(a  —  a')  =  0  ;  Avhence,  if  the  particles  are  equal,  the  sum  of 
the  vectors  from  the  center  of  gravity  to  each  ixirticle  is  zero;  and, 
if  unequal,  and  the  length  of  each  vector  is  increased  propor- 
tionately to  the  weight  of  each  particle,  their  sum  is  zero.     For 

equal  particles,  a  =  ^^^-^^,  or  the  center  of  qravity  of  a  system  of 

2w' 
equal  particles  is  the  mean  iwint  (Art.  IS)  of  the  polyedron  of 
tohich   the  particles  are  the   vertices.     For  a  continuous   bod}' 
whose  weight  is  ?r,  volume  f ,  and  density  i>  at  the  extremity  of 

a,  a  =  "^^"'^'"- ,  in  which  2  mav  be  replaced  by  the  integral  sign 

2r)f?i' 
if  the  density  is  a  known  function  of  the  volume.     For  a  homo- 
geneous body,  a  =  — — —.,  which  is  applicable  to  lines,  surfiices 

or  solids.  V  repi-esenting  a  line,  area  or  vohune.  Thus,  for  a 
plane  curve  p  =  c^(<)  =  u,'  dv  =  ds  =  Tdp  =  T(f)'{t)dt  and 


r< 


cf,(f)Tcl,'(t)dt 


JTi>'it)cU 


226  QrATEENIONS. 

4.    General  conditions  of  equilibrium  of  a  solid   bodij.      Let 

the  forces  fi',  fS','  ,  act  at  the  pomts  a',  a','  of  a  solid  body, 

and  oa' =  a,'  ox"  =  a',' RepUxcing  each  face  b}'  an  equal 

one  at  the  origin  and  a  couple,  the  given  S3stem  will  be  equiva- 
lent to  a  system  of  concurring  forces  at  the  origin  and  a  system 
of  couples.     Hence,  for  equilibrium, 

2/3'=  0,  (d) 

%Ya'/3'=  0.  (e) 

Let   ^  be   the  vector   to    any   point   x.      Then,   from    (d), 
Y  .  12/3'=:  0,  and  therefore,  from  (e) ,  V  .  ^2^'=  2Va'/8' ;  whence 

2T/3'a'-  ^\l3'i  =  2V/3'(a'-  ^)  =  0.  (/) 

Conversely,  i  being  a  vector  to  any  point,  the  resultant  couple, 
for  equilibrium,  is  2V(a'-  |)y8'=  0  ;  .-.  2Va'^'=  0  and  2/3'=  0. 
Therefore  (./')  is  the  necessary  and  sufficient  condition  of  equi- 
librium. 

This  condition  may  be  otherwise  expressed  by  the  principle 

of  virtual  moments.     Let  8,  8','  be  the  displacements.     Then 

the  virtual  moment  of  /3'  is  —  S^'8' ;  and,  for  equilibrium, 
2S;8'S'=0.  This  equation  involves  (d)  and  (e).  Tims,  if  the 
displacement  corresponds  to  a  simple  translation,  8' =8"  =8'" 
=  etc.  =  «  constant,  and  we  ma}'  write  2S3'3' =  S3S/3' =  0  ; 
whence,  since  8  is  real,  2^8'=  0.  Again,  if  the  displacement 
corresponds  to  a  rotation  about  an  axis  ^,  C  being  a  unit  vector 
along  the  axis, 

a'=  C-Ka'  =  t\^:a'-i-  T^a')  =  -  (Ka' -  D^^^! 

the  last  term  being  a  vector  perpendicular  to  the  axis.  For  a 
rotation  about  this  axis  tln-ough  an  angle  6,  this  term  becomes 

—  Z;V^4a'=  —  C cos 6'  V^a'+  HuiOY'Ca',  and  a'  becomes 

a'i=  -  CSta'-  C  COS^  VCa'+  sin^  Y.'a,' 

which,  for  an  infinitely  small  displacement, 


APPLICATIONS   TO   LOCI.  227 

Placing  tho  scalar  factor  under  the  vector  sign   and  writing  C 
simply  for  6^,  to  denote;  the  indefinitel}'  short  vector  along  oz, 

a'+S'=a'+y^a'; 

or,  8'=  VCa!     Hence  SS/8'8'=  2Si3'V^a'=  St2Va'/3' ;  or,  since  'C  is 
not  zero,  2 Va '/:?'=  0. 

5.    Illustrations. 

(1)  Three  concurrent  forces,  represented  in  magnitude  and 
direction  by  the  mediaJs  of  any  triangle,  are  in  equilibrium. 
(See  Ex.  2,  Art.  18.) 

(2)  If  three  concurring  forces  are  in  equililjriuni,  they  are 
complanar.  By  condition,  /5'+/J"4-;8'"=  U.  Operating  with 
S  .  ^'^"  X  ,  we  have  S/?'^"/?"'=  0. 

(3)  In  the  preceding  case,  operating  with  V.  /3'x,  we  have 
Yf3'(3"-\-\f3'f3"'=0 ;  whence,  since  the  forces  are  complanar, 
TV/8'/?"=  T\(3'fr,"  or  b'b"  sin(/S;  /5")  =  b'b'"  sin(^;  ^"') .  A  sim- 
ilar relation  may  be  found  for  any  two  of  the  forces  ;  whence 

b' :  b"  :  b'" :  :  sin(^;'  13'") :  sin  (^;  13'") :  sin {(3',  f3")  . 

(4)  If  two  foi-ces  are  represented  in  magnitude  and  position 
by  two  chords  of  a  semicircle  drawn  from  a  point  on  the  circum- 
ference, the  diameter  through  the  point  represents  the  resultant. 

(5)  A  weight,  iv',  rests  on  the  arc  of  a  vertical  plane  curve, 
and  is  connected,  by  a  cord  passing  over  a  pulley,  with  another 
weight,  ivl'  Find  the  relation  between  the  weights  for  equili- 
brium. 

(a)  Let  the  curve  be  a  parabola,  and  the  pulley  at  the  focus. 
Then,  froni  Eq.  (a)  of  this  article,  the  equation  of  the  curve  be- 
ing p  =  -^  {>J--l>')<^-\-Ul^'  '^ve  have 


.(|„^,)(_..V'^%0")  =  0, 


228  QUATERNIONS, 

in  which  )•  =  radius  vector.     Heuce 

or,  since  r  =  .r  +  j^,  ^v'—^o'.'     Heuce,  if  the  weights  are  equal, 
equilibrium  will  exist  at  all  points  of  the  curve. 

(b)  Let  the  curve  be  a  circle  and  the  pulley  at  a  distance  m 
from  the  curve  on  the  vertical  diameter  produced.  With  the 
origin  at  the  highest  point  of  the  circle,  p  =  xa  +  y'Znx  —  x^{3. 
Heuce,  r  being  the  distance  of  the  pulley  from  ic', 


R  +  «i 


(c)  Let  ic  he  placed  on  the  concave  are  of  a  vertical  circle, 
and  acted  upon  by  a  repulsive  force  varying  inversely  as  the 
square  of  the  distance  from  the  lowest  point  of  the  circle.  To 
find  the  position  of  equilibrium.  The  origin  being  at  the  lowest 
point  of  the  circle,  and  r  the  distance  required,  let  2^  ^e  the 
intensity  of  the  force  at  a  unit's  distance  ;  then  4,  will  be  its 
intensity  for  an^'  distance  >',  and 

whence  , — 

r  =  \  — 

(d)  Let  w'  rest  on  a  right  line  inclined  at  an  angle  0  to  the 
horizontal,  and  connected  with  w"  b^-  a  cord  passing  over  a  pul- 
ley at  the  upper  end  of  the  line.  Find  the  relation  between  the 
weights.  With  the  origin  at  the  lower  end  of  the  line,  its  equa- 
tion is  p  =  xa.  If  /S  is  in  the  direction  of  iv',  then  Sa(iv'ft-\-ic"a) 
=  0;     .-.   w"=  iv' sinO. 

(G)  To  find  the  center  of  gravity  of  three  equal  particles  at 
the  vertices  of  a  triangle,     a,  b,  c  being  the  vertices,  the  vector 


APPLICATIONS    TO    LOCI.  229 

from  A  to  tlio  ('(Mitri-  of  ij,-nivitv  of  the  weights  at  A  and  b  is 
4ai!  =  ai).  The  veetor  to  the  eenter  of  gravity  of  the  three 
weights  is  |(An  +  ac)  =  ^au  -f  .rnc  =  ^ab  4-.t(—  |^ab  +  ac)  ; 
.-.  x—j^,  and  tlie  required  point  is  the  center  of  gravity  of  the 
triangle. 

(7)   Find  the  center  of  gravity  of  the  perimeter  of  a  triangle. 

(.S)  Find  the  center  of  graA'ity  of  four  ecjual  particles  at  the 
vertices  of  a  tctraedron. 

(D)  Sliow  th;it  the  center  of  gravity  of  four  equal  particles 
at  the  angular  points  of  an}-  quadrilateral  is  at  the  middle  point 
of  the  line  joining  the  middle  points  of  a  pair  of  opposite  sides. 

(10)  The  center  of  gravity  of  the  triangle  formed  by  joining 
the  extremities  of  perpendiculai's,  erected  outwards,  at  the  mid- 
dle points  of  any  triangle,  and  ])roportional  to  the  corresponding 
sides,  coincides  witli  that  of  the  original  triangle.  Let  abc  be 
the  triangle,  va:  =  2  a,  ca  — '2  f3  and  e  a  vector  perpendicular  to 
the  i)lane  of  the  triangle.  Then,  if  m  is  the  given  ratio,  b  the 
initial  point,  and  K,,  k,,  K3  the  extremities  of  the  perpendiculars 
to  iJC,  ca,  ab,  respectively, 

BUi  =  a  +  ?;iea,     Ull.^  =  2  a -{- fS -\- Dlefi,     BK..  =  a  + /3  —  »(€(a +/5)  ; 
.-.    ^(BKi  +  BB,  +  BB,)  -  i(4  a  +  2/3)  -  i[-2a  +  2(a  +  fS)]. 

(11)  To  fnid  tlic  center  of  gravity  of  a  circular  arc.  The 
equation  of  the  ciix-k'  p  =  r(cos^  .  u  +  sin^  .  ^),  gives  dp  = 
r{-»mO  .  a  +  cos^  .  f3)(W; 


,-.   a,  = 


j  (f>{e)T<f>'{0)d6       P(cos^  .  a  +  sin  $  .  /3)dd 

j''rcf>'{0)dO  Cdo 


For  an  arc  of  90°    integrating  between  the  limits   -    and  0, 
ttj  =  — (a  -(-  /5) ,  the  distance  from  the  center  being  —  V2  ;  wliich 


230  QUATERNIONS. 

may  be  olitained  directly-  also  by  integrating  between  the  limits 
-  and .     For  a  semicircumference  or  arc  of  60°  we  have,  in 

v,  '^     2r       -,  3r 

hke  manner,  —  and  — 

TT  TT 

(12)  If  a,  yS,  y  are  the  vector  edges  of  any  tetraedron,  the 
origin  being  at  the  vertex,  then  p  —  a,  ^  —  y,  a  —  /3  are  lines  of 
the  base,  p  being  any  vector  to  its  plane.  Hence  this  plane  is 
represented  by  S  (p  —  a)  (^8  -  y)  (a  -  /3)  =  0  ;  .-.  Sp  {YafS  + 
Vya  +  T/3y)  —  Sa/3y  =  0.  If  8  be  the  vector  perpendicular  on 
the  base, 

and,  taking  the  tensors, 

T(ya/3  +  \/3y  +  Vya)  =  ^LAJ^  =  2  area  base, 
alt. 

But  Va/3  +  V/3y  +  Vya  +  Vy8a  +  Vy/?  +  yay  =  0,  in  which  ths 
last  terms  are  twice  the  wc/o?- areas  of  the  plane  faces.  The 
sum  of  the  vector  ai'eas  of  all  the  faces  is  therefore  zero.  Since 
any  polyedron  ma}'  be  divided  into  tetraedra  by  plane  sections, 
whose  vector  areas  will  have  the  same  numerical  coefficient,  but 
have  opposite  signs  two  and  two,  the  sum  of  the  vector  areas  of 
any  polyedroyi  is  zero.  These  vector  areas  represent  the  pres- 
sures on  the  faces  of  a  polyedron  immersed  in  a  perfect  fluid 
subjected  to  no  external  forces.  For  rotation,  since  the  points 
of  application  of  these  pressures  are  the  centers  of  gravity  of 
the  faces,  to  Avhich  the  vectors  are 

i(a  +  /3  +  y),      i(^  +  a),      i(y  +  ^),      i(a  +  y), 

we  have  the  couples 

-iVKa+^  +  y)(Va^+V/8y  +  Yya)+(a  +  ^)V/?a+(^  +  y) 
Vy^  +  (y  +  a)Vay| 
=  _iV(aV/?y  +  /3Vya  +  yVa/3), 

since  aVu^S  +  aV/3a  =  0,  etc.     But,  Equation  (123),  this  sum  is 
zero.     Hence  there  is  no  rotation. 


i    UIVJVER8ITY 


LD21. 


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hfRKrifv 


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